I’ve also worked as a consultant. In that capacity, I visited many math departments, and offered many professional development workshops to colleagues around the country. One thing I learned was that many, many math departments are home to teachers with seriously divergent philosophies. While we cannot ignore teachers’ need for agency, we should also think about how these differences impact student learning. In this post, I will argue that striving for departmental consensus is well worth the required time and effort, even if the process is slow and challenging.

Philosophical differences in a department can be generational. A young teacher, fresh out of college, may join a department of oldsters who have always done it a certain way, and have no intention of making changes. They may even require the new hire to ignore what they learned in Ed School, and do everything their way. I will not discuss that situation here. Retirements and the influx of new colleagues can help.

A more common problem is philosophical tensions among well-intentioned and open-minded colleagues. This can look like a manifestation of the so-called math wars within a department: direct instruction vs. student intellectual engagement, skills vs. concepts, memorization vs. understanding, etc. In my view, these binaries are profoundly misleading. Direct instruction works best on a foundation of student intellectual engagement. Concepts cannot be acquired in the absence of skills, and vice versa. Memorization can help if it encapsulates understanding — not if it substitutes for it. Working for a departmental consensus is one way to work on embracing both sides of these false contraries and developing a local version of a balanced approach.

Every situation is different, but here is a possible scheme to get there.

1. Reach agreement on scope and sequence: what should be included in each course? I realize that you may not have a choice on this because of district or state constraints. If that is the case, that initial discussion should be about what should be emphasized in each course. The reason this has to come first is that without such agreement, resentments can develop (“They should have learned this last year!”) I share some big-picture planning ideas in this article, which can help structure that conversation.

2. Have teachers with different backgrounds, views, and experience teach different sections of each core course, in parallel. This should involve regular meetings to debrief the previous week and map out the next week, as well as the recording of notes of how things turn out. This is a slow but profound way for teachers to learn from each other as they share strategies in a fine-grained way throughout the course. I share ideas on teacher collaboration in Chapter 12 of There Is No One Way to Teach Math. (In fact, the book should help with the consensus-seeking process: each chapter ends with discussion questions which can help develop the department into a professional learning community.)

3. For maximum impact, do not let teachers over-specialize. (“I teach Algebra 2.”) After a few years, teachers should move on to different courses in the core sequence. Over time that will lead to joint ownership of the whole program. (This mobility is mostly needed in the Algebra 1/Math 1 to Precalculus sequence. Elective courses are experienced by fewer students and need not be part of the rotation.)

4. Take a few days each summer to make needed adjustments to the scope and sequence, and (probably more important) to introduce representations and learning tools that could enhance student learning of the most important ideas. (Administrators: make sure teachers get paid for this important summer work!)

Over time this process will yield a department that is more than the sum of its teachers. Students will appreciate the consistency and lose interest in comparing one teacher to another. Most importantly, their learning will be enhanced, as they will benefit from the combined contributions of the whole department, not just the quirks of their current teacher.

That said, it’s important that each teacher maintain their own individuality in the midst of this joint project. If someone is a history of math buff, or is enthusiastic about a relevant “enrichment” topic, or wants to make connections with social justice — by all means they can try to squeeze that in, as long as they prioritize implementing departmental policies.

Amanda worked briefly at my school many years ago. Later, we collaborated on expanding a couple of lessons of mine into full-on lesson plans (McNuggets, and Cell Phone Plans.) Yet later, she contributed an insightful comment to my blog post Freakonomics Radio on Math Curriculum. Here is what she has to say about the two books.

— Henri

Two Math Education Books

By Amanda Cangelosi (amanda.cangelosi at usu dot edu)

For the past decade, I’ve spent half of my summers facilitating a methods course for pre- and in-service teachers who are seeking to attach a secondary math endorsement to their license in Utah.  As it would be unthinkable to fail to incorporate Peter Liljedahl’s Building Thinking Classrooms (BTC) into this course, I make “The Orange Bible” required reading.  The 14 practices described in BTC are essential and warmly-welcomed by my teacher-students (and adored by me, possibly to the point of worship).  The other required text is Picciotto and Pemantle’s There is No One Way to Teach Math (N1W), which serves as a comprehensive, validating, and uplifting resource for an audience that is overwhelmed by the nature of their careers.  My course assigns N1W to be read first, followed by BTC second.  Reading N1W first is helpful for several reasons, one of which is that nobody likes to be told what to do.

My course is not exactly voluntary for my teacher-students.  State mandates require that those who want to teach secondary math must earn an endorsement, and while there are a handful of pathways for the endorsement to be acquired, taking a 7-week summer course for college credit via Zoom is often the simplest route.  Thus, the experience isn’t quite the same as a district-sponsored book club; some people enter the virtual classroom understandably annoyed and skeptical.  Are these folks going to endure yet another out-of-touch teacher educator sellout talking at them about the latest fads?  The answer would be “yes” if not for N1W.  The immediate validation that comes from the book title alone is enough to open hearts and minds, which is a great place to begin a discussion-based class.  

Teacher validation is overlooked as a necessary step in inviting large-scale change.  We have to meet people where they are, which we seem to understand for children, forgetting that adults are just children who are older.  To make complex modes of instruction accessible, we can’t just explain and justify the modes; we need to first say, “I see you, and you bring value to this space,” because that’s the truth.  As N1W tells us in the Special Note to Young Teachers, “You are who you are. That is a crucial gift to your students.” When we transition to reading BTC, instead of being overwhelmed by the 14 practices, teachers appreciate remembering that “it cannot be rushed.”   

Since our identities include our politics, and since politics intersect with the Math Wars, an important facet of teacher validation is addressing the Math Wars problem.  N1W nips the Math Wars problem in the bud, rescuing my class from unproductive debates and false dichotomies.  By embracing contraries, N1W fosters a spirit of camaraderie, setting the stage for growth, as opposed to a futile pursuit of “what works.”  When teachers hear that whatever instructional modes they are currently practicing have a place, they are being told that they are whole instead of broken. Perhaps ironically, this validation allows teachers to move forward. The polarization spell is broken.  Teachers are ready to grow their collection of practices, adding to their instructional repertoire and seeing value in each component. 

One powerful excerpt from N1W that clicked with my teacher-students, and dovetails nicely with BTC:  

“All of the above”, but not “anything goes!”  Our foundational idea is that math teachers should reject shallow either-or stances, and instead learn to combine multiple approaches, even if they superficially seem to contradict each other.  There is no one way: we need an eclectic mix of techniques that prioritize student understanding. . . . The main path to student understanding is intellectual engagement [i.e., thinking, as in a Thinking Classroom]. . . . the main vehicle for that engagement: problem solving [e.g., non-curricular tasks].

When my class reads about non-curricular tasks in BTC, it is helpful to have already read about puzzles in N1W.  From N1W, we learn that a puzzle is “a relationship between the puzzle constructor and the puzzle solver” and that it must be solvable, fair, challenging, and the solution must be satisfying.  Puzzles “help lower the emotional stakes while offering a path to engagement for a wider range of students.” And N1W continues to dive deeper into the characteristics of a good puzzle, providing examples.  This made Chapter 1 of BTC a breeze, with minimal panic about non-curricular endeavors.  

Understanding the value of eclecticism from N1W, ahead of reading BTC, is fruitful. We must remember that research is a thing because variability exists. It is soothing for my teacher-students to hear—from a published source—that just because a study concludes that certain practices are optimal doesn’t mean it’s what they, as individuals, should be doing (and certainly not all the time).  Rather, the study provides really good information for them to add to their arsenal.  Once any given practice becomes mass-normalized, it may lose its power; the key is to have an ever-growing collection of ideas at your disposal to shake things up. 

I admit that I’m guilty of rolling my eyes and disengaging at professional development meetings whenever the giant post-its, accompanied by smaller post-its and markers come out.  The post-its-atop-post-its got old, and I don’t want to play that game anymore, because too much routine begins to feel patronizing after a while.  According to N1W, “It is a mistake to put all one’s pedagogical eggs in one basket. . . . Students do appreciate some predictable routines, but too much of that can take the life out of a class.  Switching modes can contribute to welcome changes of pace, and bring some variety to the program.”  When my teacher-students know this, they are more open to taking risks, because they don’t feel locked-in to playing one game all year.   

There are five comprehensive chapters in N1W that address modes of instruction, including many examples of multiple representations and manipulatives. Each of these can be incorporated in BTC-style tasks, and my teacher-students appreciate having seen so many examples when they read The Orange Bible.  In addition to fleshed-out examples of Thinking Classroom tasks, N1W addresses homework, planning, assessment, teacher joy, philosophical underpinnings, and professional development.  

Regarding teacher joy and self-care, an important intersection of BTC and N1W is that of giving teachers permission to teach in ways that are consistent with their values.  In BTC, Liljedahl says, “We need to start evaluating what we value,” where what we value is not only content knowledge but, crucially, habits of mind such as perseverance and collaboration.  In N1W, Picciotto and Pemantle include strategies for “participation quizzes,” and further invite teachers to, more generally, “…trust your intuition, avoid dogma, be flexible, be kind. . . . even if your intuition is `wrong’, at least it is yours.  Your students deserve to get the real you, not a poor imitation.”

“The real you,” as a teacher, is more than the tasks you choose, the way you arrange your classroom, the modes you choose to facilitate activities, and the way you evaluate students.  The fullness of N1W helps my teacher-students in ways beyond what the main title suggests: It is packed with concrete, usable ideas that reflect the lived experiences of the authors.  It is comforting to read a book that is non-academic, personal, and steeped in the wisdom built from decades of the authors’ classroom experience.  Let us consider the subtitles of BTC and N1W, respectively:  “14 Teaching Practices for Enhancing Learning” and “Actionable Ideas for Grades 6-12.”  I don’t know about you, but to me the word “practices” signals something research-based (e.g., 5 Practices; Standards for Mathematical Practice; Principles to Actions’s eight  “Effective Mathematics Teaching Practices”), while the gentle word “ideas” feels personal and anecdotal, signaling a conversation with trusted colleagues.  Obviously, both are important, which is why utilizing both books is wonderful.  

I shall attempt to wrap things up by way of a poor metaphor.  When I was a child, I attended many birthday parties at Aladdin’s Castle, an arcade at my local mall.  The best game was Ms. Pac-Man (duh), so I played that the most.  But other games were great, too; I wasn’t about to put all of my tokens in one machine, no matter how much I loved Ms. Pac-Man on a spiritual level.  There are a lot of great games to play as a math teacher.  You can play Building Thinking Classrooms (Ms. Pac-Man), 5 Practices (Pac-Man), Complex Instruction (Mega Man), and so on.  If instructional frameworks are games to play, then There is No One Way to Teach Math pours you a hot coffee, double-checks your seat belt, drives you to the arcade, suggests winning strategies, reminds you that you don’t have to play any of the games, and then ensures you make it home safely.

Teaching is more complex than playing a game, even when the game is complex; teaching demands the teacher’s whole self.  In an alarmist culture of math education, it’s important to have cutting-edge research-based resources like BTC and also a comprehensive home base from experienced colleagues that validates and uplifts our individuality and supplies a wide range of tools and activities, such as N1W.  I enthusiastically utilize both.

I had the good fortune of decent working conditions, which allowed me to teach for 43 years at almost all K-12 levels. In 2013, I retired and joined the ranks of the consultants and curriculum developers who try to help from the outside. I’ve been able to do this reasonably well, because all these years of teaching experience allowed me to grow both mathematically and pedagogically, meaning that I do have some hard-earned wisdom (and curriculum materials) to share — as you will see in this post.

An Easy Calculation Can Mask an Important Concept

I recently read Dan Meyer’s insightful piece “Why Teaching is Harder When the Math is Easier”. He points out that calculating slope is a straightforward proposition, but that this very fact makes it difficult to get across all the concepts that are embedded in that simple calculation. Typically, we teach “rise over run” to students, and keep reminding them about this. (One French educator told me “rise over run is a religion for American math teachers.”)

Because the underlying concept is so important, I had to face this particular challenge head-on in my decades of teaching high school students about slope. You can find some of the resulting curriculum materials on this page of my website: Rate of Change. In today’s post, I’ll try to share the pedagogy behind those activities. Note that “slope” is about the representation of rate of change on a graph. Rate of change is the underlying concept across representations, which explains the title of this blog post and of the corresponding page on my website.

Before getting into the specifics, I’d like to make a big-picture observation: If a concept is important, the goal should be to find multiple ways to teach it. Searching for “the best” way to teach it is a fool’s errand. Yes, some approaches are better than others, but nothing works as well as teaching the concept in more than one way. I discussed this before on this blog:

• In 2015, in How To, I critiqued the idea that there is one best way to teach any given topic, using equation solving as an example.
• In 2016, I challenged those who believe they have found the way to teach math in Eclectic and the subsequent posts.
• In 2020, I wrote No One Way, which I used to explain my website’s motto (“There is no one way.”) Using examples spanning all grade levels, I argued that it is the math itself that demands that we approach important topics in multiple ways. 

Much of the above ended up in my recent book about math pedagogy, (co-authored with Robin Pemantle). This is in fact reflected in the book’s title: There Is No One Way to Teach Math: Actionable Ideas for Grades 6-12. Read about it here, see the Table of Contents here, and order it here. Until 31 March 2026, you can get 20% off.

On to the specific pedagogical insights embedded in my various rate of change activities.

No Three on a Line

How many points can you put on an n by n lattice so that no three points are on a line? There is a conjecture that you can place 2n points under this constraint, but as n gets bigger, it’s not so easy to do. Here is a successful example where n = 4:

Having students check each other’s attempts forces a discussion of slope — possibly long before the concept has been discussed or even named. This grounds the algebraic concept in a concrete challenge. This activity can be done in middle school, or even  late elementary school. Or frankly at any time: the challenge is equally engaging for students at any level and even to their teachers.

Pedagogical insights:

• A hands-on puzzle with no technical jargon can trigger curiosity, engagement, and a foundation for discussion.
• Discrete explorations on a lattice provide a foundation for later work in the Cartesian plane.

Link: No Three on a Line

Similar Rectangles

Two rectangles are similar if their sides are in a proportional relationship. This situation is simpler than what we have for triangles, or for figures in general, because similarity requires equal angles. That, of course, is a given in rectangles since all the angles are right angles. A visual way to assess whether sides are proportional is to superpose the rectangles, in a way that they share a vertex, and the short and long sides extend in the same direction. In this setup, if they share a diagonal, then the rectangles are indeed similar. This is the basis of a lesson in Geometry Labs (section 10.2, possibly preceded by 10.1), which opens with this figure:

This diagonal test rests on understanding slope: if the rectangles share a diagonal, it means that the “rise over run” for each are equivalent fractions. 

That same lesson ends with hands-on geoboard challenges: find families of similar rectangles on the 11 by 11 geoboard, and find geoboard slopes between 1 and 2. 

In a geometry class, the lesson can be used to review or preview similarity — and/or review slope. In an algebra class, it provides a concrete geometric foundation for slope.

Pedagogical insights:

• Algebra-geometry connections reinforce understanding in both domains.
• Searching for examples within mathematical constraints is an engaging group-worthy activity which provides repeated opportunities to discuss and clarify the constraints.
• Discrete explorations on a lattice provide a foundation for later work in the Cartesian plane.

Link: Similar Rectangles (scroll down)

Algebra: Themes, Tools, Concepts

Algebra: Themes, Tools, Concepts is the textbook I co-authored with Anita Wah in the 1990’s. Our commitment to “no one way” went along with a belief in a spiraling curriculum, as illustrated by photos at the start of each chapter. The importance of slope is reflected in the fact that we included that topic, more or less explicitly, in at least two dozen lessons. Those involve function diagrams (see below), input-output tables, Cartesian graphs, and “real world” examples.

Links to examples of the latter type:

• A preview involving Celsius-Fahrenheit conversion: A Hot Day
• Density of a mystery substance: In the Lab
• Graphing circumference vs. diameter for jars in Jarring Discoveries (scroll down)
• A growing child’s age, height, and weight.
• Stair safety: Stairs and Squares
• Modeling Motion

Pedagogical insight:

• “Real world” examples are relatable, and offer a context to reinforce the use of numerical and graphical representations.

Link: Algebra: Themes, Tools, Concepts

Function Diagrams

Function diagrams are a parallel axes representation of functions. For example, here is a diagram for y = 2x – 3:

In this representation, you can think of rate of change as magnification. In the above example, when x increases by (say) 3 units, y increases by 6 units: the magnification is 2. I try to use the word slope when discussing a Cartesian graph, rate of change when discussing a table of values, and magnification when discussing a function diagram. I have a fond memory of a student exclaiming: “Those are all the same thing!”, something I do not initially stress, or even mention. 

An advanced use of function diagrams is making an intuitive case for the chain rule in calculus.

Pedagogical insights:

• An unfamiliar representation can provide a fresh context to talk about familiar concepts, thus helping the development of deeper understanding, evening the playing field among students, and triggering interest among those who would resent the repetition of same-old approaches.
• Some representations can be used over a wide range of grade levels.

Link: Function Diagrams

Slope Triangles

The visual representation of “rise over run” is a slope triangle. Adjacent slope triangles make a sort of visual staircase. I took advantage of that by creating a puzzle-like GeoGebra-based applet called Stairs. Here is an image that should give you a sense of how it works:

Lines can be graphed in there as well, which allows for additional possibilities, such as point-point or point-slope graphing puzzles.

I do not force specific lessons using the applet. It can be used with the exercises and challenges in the accompanying worksheets, or with problems posed by students or teachers.

Pedagogical insights:

• A hands-on puzzle can trigger curiosity, engagement, and a foundation for discussion.
• A visual representation of “rise” and “run” helps anchor the “rise over run” mantra in an intuitive way..
• Mistakes are instructive, and pave the way for correct solutions.

Link: Stairs

Make These Designs

A standard but ineffective use of electronic graphers is “graph this, graph that, what do you notice?” The problem with that approach is that the teacher (or the worksheet) is doing all the heavy lifting by suggesting what to graph. If instead students are given designs to make using y = mx + b, the tables are turned: it is the students who have to make the important decisions: choosing the m and b that will achieve the desired result. Here is a typical target design from the graphing-calculator era:

The activity can be done in any electronic grapher. It is forgiving: as in the Stairs applet, even unintended results can yield some learning. One way to reinforce understanding is to ask students to write explanations of how they created each design.

Pedagogical insights:

• A hands-on puzzle can trigger curiosity, engagement, and a foundation for discussion.
• “Mistakes” are instructive, and pave the way for correct solutions.
• “Low threshold, no ceiling” activities work with a wide range of students.
• Writing helps to reinforce understanding.

Link: Make These Designs

Doctor Dimension

This is a geometric context to discuss rate of change in piecewise functions. The activities range from eighth to twelfth grade. The basic idea is to see how the rate of change changes as the “liquid” fills the “containers” as displayed above. This includes linear and nonlinear cases, and the inspection of the corresponding graphs. In precalculus, it allows for a discussion of “concave up” vs. “concave down”. In calculus, continuity and differentiability get into the picture. The unit culminates with a reversal: students design containers to match given graphs.

Pedagogical insights:

• A hands-on puzzle can trigger curiosity, engagement, and a foundation for discussion.
• Algebra-geometry connections reinforce understanding in both domains.
• A rich context can yield insights into multiple concepts.

Link: Doctor Dimension

Slope Angles

Different slopes yield different angles between the line and the x-axis. Acknowledging this bit of geometry in the Cartesian plane yields an unusual but very effective introduction to trig ratios. We can use students’ reasonably solid understanding of slope as a different way to introduce the tangent ratio: “rise over run” is a particular case of “opposite over adjacent”. So the standard trig exercise suggested by this image can be solved without mentioning trig at all:

This involves the use of the ten-centimeter circle, which allows students to find the slope corresponding to 39° using a ruler. Once the slope is known, calculating the length of the flagpole is straightforward: the run is given, and the number we seek is the rise. Once this is understood, the tangent ratio can be introduced, along with the corresponding calculator key or other electronic method to find the tangent or arctan.

This approach is based on student understanding, unlike the usual “sohcahtoa” mantra which relies on memorization. After that, I have found it useful to delay the introduction of the sine and cosine. This creates a later opportunity to review the tangent ratio, and to finally introduce “sohcahtoa” or “soppy cadjy toad”.

Pedagogical insights:

• Algebra-geometry connections can provide a powerful way into trigonometry.
• Vocabulary and notation are best introduced when there is some underlying understanding.
• Traditional sequencing of topics is not necessarily the best option.

Links: Slope Angles and Geometry Labs

Pattern Block Trains

This activity originated as a fifth grade introduction to rate of change, but I saved it for last because I extended it to end with an extremely challenging question, pretty much aimed at teachers. 

Pedagogical insights:

• A hands-on activity can trigger curiosity, engagement, and a foundation for discussion.
• Algebra-geometry connections can yield activities at many grade levels
• Activities intended for students can yield interesting mathematical questions for teachers.

Link: Pattern Block Trains

Formal Proof

A curious fact about high school math is that students (and teachers!) never see a proof that the graph of y = mx + b is indeed a straight line, and conversely that a non-vertical line satisfies an equation of the form y = mx + b. I offer a worksheet that guides teachers through such a proof. Time constraints have prevented me from trying this with students, but the prerequisites are basic geometry theorems and some mathematical maturity.

Link: The Geometry of y = mx + b

Conclusion

Teaching the same thing the same way will yield the same result. Understanding an idea in multiple ways is an indicator of expertise. A quick scan of the above shows images that look completely different from each other. This is a further pedagogical insight: an approach depending on multiple representations and multiple contexts has substantial benefits.

• For some students, it offers repeated entryways into the concept. This is crucial if the initial lessons do not break through or sink in.
• For students who grasp the concept early, it avoids boredom and presents varied applications of the concept. This is far preferable to “review” that consists of a rehash of the same lesson. 
• It keeps the concept alive over an extended period of time (in this case, all secondary school grade levels.)

Students appreciate variety. Math class should not always be the same day after day! One way to mix it up, as you see with the examples above is to use different learning tools: dot paper, geoboard, input-output tables, electronic graphers, online applets, function diagrams, the ten-centimeter circle, pattern blocks…

Well, I rest my case. I hope that I made my point, or at least that you found some activities to diversify and enrich your teaching!

— Henri

— Henri

Algebra and Geometry

— Henri

Mentoring and Coaching Teachers

by Margot Schou

I began my teaching career, fresh out of college, at the Urban School of San Francisco. How lucky was I to have Henri Picciotto as my first mentor? Nineteen years later, I continue to use his wisdom in the classroom. More recently, as I have had the opportunity to mentor and coach new teachers, his words have informed my work in a new way. I am excited to be a guest contributor to his blog! 

No One Way

One of my favorite Henri-isms is: “there is no one way”. Teachers make myriad decisions for each class, and after every observation I like to ask about the choices the teacher made. “Why did you do it that way?” There is no wrong or right answer, but it is important to think about why we are making each decision. For example, did we want the students to understand how to do that proof, or do we just want them to have some chance of remembering the resulting theorem or formula?

For example, if I want students to be able to explain the formula for the sum of arithmetic series, I’ll ask them to repeat the process of copying the series, flipping the numbers and placing them on top of the original, adding, multiplying, solving several times with numbers before deriving the formula more generally [i]. For something like the product rule for differentiation, however, I don’t need them to memorize the derivation, but doing it once together in class gives them some understanding of where the formula came from, which hopefully helps them to remember it accurately.

So, I ask my mentees, what was your goal? Was the aim to increase collaboration, or for each individual student to self-assess? Did it work? Were we trying to get more participation or more reflection? How could we do this differently in order to achieve one or the other?  I love mentoring because I always learn something about my own teaching when I observe others and ask these questions. It offers me such a great opportunity to reflect on my own decisions and to consider other ways.

In addition to “no one way” pertaining to our teaching goals, I also like to think of “no one way” as it relates to our own personal and unique teaching styles. To me, one of the most exciting things about teaching is that what works for me as a teacher might not work for you, just as what works for one student might not work for another. A big part of my job as a mentor is to help cultivate a teacher’s unique strengths. Teaching is such a relational profession: students will see right through us if we are trying to be someone we are not [ii], and students benefit from learning from a variety of personalities. Working closely with a teacher in a mentorship role allows me the opportunity to provide feedback that is specific to that teacher. Rather than generalized pedagogical approaches, the teacher and I can be creative about strategies that might work best for them.

My recent experience mentoring another math teacher exemplifies how important it is to teach to our own strengths and styles. I tend to rely on a variety of tried-and-true tactics depending on what is happening in the class in front of me – show me the answer on your fingers, give me a thumbs up or thumbs down, tell your neighbor and then raise your hand [iii] – rather than planning out in advance how I will encourage participation from my students. My mentee preferred a more structured approach. Another colleague shared a video of a teacher giving their students a “participation quiz” and my mentee and I discussed how she might incorporate this tool into her classroom. She made a version of the quiz that felt like her own, and it was a success. Especially as we aim to recruit, hire and sustain a more diverse faculty, we need to be sure that we are supporting and encouraging the perspectives and teaching that they offer. How are we setting all of our new teachers up for success?

Excitement

After observing one of my classes, in my very early days of teaching, Henri said to me: “Change the tone of your voice when something is important. Be excited.” As a trained performer, Henri taught me strategies like changing the tone of your voice, going silent, typing out and projecting on the screen your observations of the class. All of these methods can highlight important topics, concepts, procedures, and habits. I think of this advice nearly every day, and I regularly share it with new teachers. Quite often, teaching is a performance. I sometimes dance in class, and I regularly sing. Find the performance piece that works for you.

While Henri meant it as an act, in my own practice I have also learned the power of sharing my genuine excitement with my students. Sometimes, when we pretend to be excited about a lesson, we actually become excited. I have taught Algebra I more times than I’d like to count, so when I say “check it out this is so cool” when completing the square with students, I am not really being one hundred percent sincere until after I say it and remember, right, this is actually really cool! I teach math because I am passionate about the subject. I think that math is beautiful in and of itself, and I get to share that with my students. My hope in mentoring new teachers is that they will also love and be excited by teaching.

Coaching and collaborating with more seasoned teachers provides a unique opportunity to bring the excitement back! We can discuss various pedagogical approaches, try new things together, and learn from each other. I recently had the opportunity to observe an experienced Spanish teacher who was new to our school as part of a regular onboarding process. Being in his class reminded me of what it was like to be a student, the amount of processing time required and the energy needed to keep up. I could translate some of my experiences in math classes and offer this teacher different strategies for their classroom. This kind of collaboration allows us to always grow, and it offers a new perspective and opportunity to improve.

Make it sustainable

The first year at a new school can be overwhelming and exhausting. My role as a mentor is to help minimize the overwhelm, to show teachers that they are valued, and to support them in whatever way will help them stay excited about their job. This might mean helping to brainstorm a lesson plan, or observing a class and helping to problem solve a classroom management issue. It might mean encouraging them to go off on a mathematical tangent in class if it is something they and the students are interested in. I want to help them hold onto the joy.

I often remember when Henri said to me: “the time you spend on grading is spent on one student. The time you spend on planning is spent on a whole classroom of students.” How do we make teaching sustainable? For sure, being excited and joyful is a big part of that, but so is our time. In my experience, teachers like to talk a lot about how much work they do, how late they stayed up grading those tests or how much of their Sunday was spent planning for their week. As a new teacher, it can feel like if you aren’t working 24/7 then you aren’t doing enough. As a mentor, I try to debunk this myth and gear these conversations towards efficiency. Sometimes, I will sit down and grade with a new teacher to show them where they can take less time. After all, we give feedback every time we answer a student’s question in class. We develop relationships with our students so they feel comfortable asking a question about their test.

I encourage teachers to assign test corrections done as homework because, rather than the teacher taking the time to explain how the student got a problem wrong, the student can do the work of figuring out their mistake and correcting it. This not only saves the teacher time, but it also helps the student fully process their error, it encourages them to actually get help when they need to rather than glossing over a teacher note on the test, and it provides the student with the opportunity to deeply learn from their mistakes [iv]. I’ll also often help new teachers with lesson planning, so they don’t have to reinvent the wheel. As one of my mentees put it, “I am going to be a much better teacher in class tomorrow if I go to bed at a reasonable hour tonight.” Our teachers will be better when they are happy and fulfilled, when they take time for themselves. In the long run, the well-being of our teachers is paramount.

Providing a teacher with a mentor could only support and strengthen that teacher’s well-being. And, a good mentor can help set a teacher up for a sustainable and successful career in education. I have been very fortunate to have great mentors, and I hope that our schools will continue to invest in mentors and coaches in order to support and sustain the next generation of teachers.

[i] See Understanding Sequences and Series for lessons that implement this approach, and for a 10-minute video that shares the underlying pedagogy.

[ii] I made a similar point in Dear Young Teacher, and I included an updated and improved version of that letter as an appendix to There is No One Way to Teach Math.

[iii] I learned these “tried-and-true tactics” and other discussion-management techniques from Project SEED in my first year of teaching. See pp. 142-146 and 157-160 of There is No One Way to Teach Math.

[iv] Test corrections are an alternative to “retakes”. I discussed them in a blog post, and on pp. 192-195 of There is No One Way to Teach Math. (In fact, Chapter 11 of that book is all about assessment.)

Thanks, Margot! As it turns out, some of the ideas I shared with you back then I had learned from my own mentors in the 1970’s! All of us are part of a profession, and our individual practice is immeasurably enriched by mentoring, coaching, and more generally by every sort of collaboration. You are an inspiring example of that.

To follow up on this post, readers might take a look at Mentoring Young Teachers. And for more Henri-isms, see Catchphrases.

— Henri

I’ve always agreed with this goal, but my approach consisted mostly of adding “Explain.” to various key problems in class work, homework, and assessments. I did this both for new material, in summary and review questions, and in assessments. Nothing wrong with that, of course, but Frank and Anya’s (and their department’s) policy of separating and explicitly labeling “Argumentation” questions is a valuable technique that could be used alongside the simple and ubiquitous “Explain”. If I was still in the classroom, theirs is a practice I would be sure to adopt.

Thank you Frank and Anya!

— Henri

Integrating Argumentation in Math Class

by Frank Cassano and Anya Sturm

The question “When are we going to use this?” seems to be inevitable in math class. While we need to keep open the path to further STEM education and careers, for most students the short answer is never! Most will not need to know logarithmic properties, nor solve trigonometric equations in their everyday lives after completing their math coursework.

This being said, there are absolutely transferable skills that students develop in math class.Similar to problem solving, our framework for integrating argumentation has been created and modified in collaboration with our entire department. This allows us to have a standard assessment and grading structure across the four years of our curriculum. 

Our students regularly practice and are assessed on argumentation in most units. Argumentation questions are labeled on unit tests, and can vary from an algebraic or geometric proof to a statement that students need to justify using any tool at their disposal (algebra, graphing, paragraph, etc). These questions may be generalized versions of practice problems, or resemble a familiar proof structure.

Students struggle with knowing “how justified” they need to be, and even pure variable manipulation is challenging for many students. Because the skill of argumentation is already difficult, we’ve learned that our “argumentation” problems are most effective when they don’t feel entirely new. Often, our assessments echo the structure or type of problems practiced in class.

Here is an example from our trig function graphing unit in Precalculus class this year:

Mid-unit Argumentation Practice (ungraded):

If h(x) is a linear function with slope m and k(x) is a sinusoidal function with period T, prove that the period of k(h(x)) is T/m.

Argumentation portion of Assessment (graded):

Given f(x)=mx+b and g(x)=asin(x-h)+k, prove that f(g(x)) has a maximum value of am+mk+b.

Here are a couple of examples of student work we received on the assessment. Both of these students earned full credit.

Here are some more examples of argumentation problems that we use in our Algebra 2 and Precalculus courses.

Essentially, this post is a case in favor of teaching and assessing mathematical proofs both in and out of the geometry curriculum. Forming a conjecture, and following an evidence-based logical path to a conclusion is one of the most transferable skills students can gain in math class. Students should be doing it regardless of whether we name it or not! The structures mentioned in this post help provide intentionality and awareness to the practice and performance of argumentation. Along with the skills of problem solving and procedural fluency, this is a skill students will need regardless of what they do once they graduate.. 

Readers of this blog know that I agree with the professional consensus on the importance of incorporating problem solving throughout math instruction. In our book There Is No One Way to Teach Math, Robin Pemantle and I dedicated a whole chapter to that, and it is a major feature of the curricular materials I share on my website.

Frank and Anya contribute concrete ideas on how to do this, a worthwhile complement to my approach. With their permission, I inserted some [comments of my own] into their article.

— Henri

INTEGRATING PROBLEM SOLVING SEAMLESSLY

by Frank Cassano and Anya Sturm

We are interested in moving students from viewing math as a set of operations to perform after the teacher has explained a precise recipe to follow, to viewing math as a subject involving problem solving and critical thinking. Over the past few years, we have experimented with different classroom techniques and curriculum structures to support our students in this endeavor. Our goal is to help them engage with the creativity that complex problems require — without creating more anxiety. In this post, we aim to provide you with an example of how we have integrated problem solving throughout our units.

These changes have been adopted by our entire department. We took significant department meeting time to write grading schemes for assessing and scaffolding problem solving throughout the students’ four years with our curriculum. We have found that the parallel structure throughout our math classes has helped students engage with the uncertainty of problem solving more than implementing these changes into one class in isolation.

In order to make the skill of problem solving explicit, we flipped the traditional order of presenting material. Rather than teaching a set of problems that require one strategy, and then graduating students to more complex problems once they have achieved necessary fluency, we start units by posing a question that requires students to connect various ideas, some of which may be new. This means that students start each unit making connections, solving problems, and finding patterns, to generate motivation for the content that will come in the unit. Along the way, students are forced to practice dealing with uncertainty in their solutions, and learn how to persevere when stuck. This mitigates (some of) the fear of failure that comes with uncertainty.

On day 1 of a unit, there isn’t an expectation that students can come to a complete answer on their own. Rather, we as a class try to see if we can come to a conclusion and then explicitly name that the work that the students just did was problem solving. All that is left to do is synthesize their findings — this is where new vocabulary, equations, and procedures get formalized.

[This is similar to what I call an anchor activity, which I sketch out in part 5 of “Big Picture Planning”, citing specific examples. And “formalized” is perhaps a better choice than “institutionalize”, the word I use in “Taking Notes vs. Doing Math” and There Is No One Way to Teach Math. The idea is that after students have wrestled with a concept using their own approach and language, it is important to bring them into the international institution of mathematics. That term originated among French math educators.]

Day 1 Example: On the first day of an Algebra II exponentials unit, this is what we gave the students:

• Roll 50 dice at each table group from a ‘bucket’ onto the floor, all at once. Remove any 6’s. Keep track of how many are left. Roll again until all the dice are gone.
• After the experiment, enter data into Desmos, either as coordinates or in a table. The first pair should be (0, 50), the numbers before you did your first roll. Try fitting an equation to the data.
• After a few minutes, enter f(x) = ab^x and make a and b sliders. Play with a and b to see if they can fit your data. You will have to explore and see how to change the slider endpoints. (Here is an example data set.)

This takes a lot longer than just giving an exponential equation with definitions and doing examples! But students had to engage for 30-40 minutes, and they didn’t think about what they were doing as “problem solving”. We named it after each group had already derived an exponential function for themselves, and we gave them vocabulary to go along with it.

[Starting in the days of the TI-83 calculator, I used a similar activity to launch an exponential functions unit. It differed from Frank and Anya’s version in that I followed the experiment with a discussion that led to a “theoretical” formula prior to doing any graphing. I stand by my “think first” approach. You can download the worksheet in Exponential Functions. See also Ripples about how some ideas, including this one, go viral.]

So, we’ve practiced problem solving at the beginning of a unit, but if our goal is for students to see problem solving and math as a set of skills that are inherently connected, we need them to continue flexing their problem-solving muscles throughout the unit. In keeping with the theme of problem solving first, we use open-ended warm-up questions throughout the unit to get students working together and critically thinking at the beginning of each class to set the tone for class.

[This is similar to Scott Farrand’s Think First policy, another idea I first encountered at Asilomar.]

We also use Peter Liljedahl’s tiered homework system (labeling homework problems as mild, medium, spicy) as a way to make different parts of math explicit to students. Mild problems target skill fluency, while spicy problems target problem-solving.

Mid-unit problem-solving example: a warm-up

(assigned before we do any notes on shifting the vertical asymptote of exponential functions):

Anya gets a hot cup of coffee from the café and brings it to class. When class starts, she measures the temperature at 125° F. Anya loves the lesson so much, she keeps forgetting it and doesn’t drink any of it for the whole class. After 40 min, she checked it and the temp had dropped to 86°. She forgot to drink it all, but even the next day the temperature was still above 75°. Determine a function for the temperature of Anya’s coffee after x minutes.

Throughout a unit, students practice problem solving in class, getting feedback from us, as well as concrete tools to use, and they get opportunities to practice the skills on homework as well. This allows us to include problem solving on an assessment without the student pushback of “this looks different from the problems in class.”

In order to engage with problem solving, students need to be working on a novel problem. On every assessment, the last problem is labeled as “problem solving,” and has one element that we have not touched on in the unit.

[This is better than my own practice of labeling such problems on tests as “bonus”. First of all, doing it on every assessment is powerful in affecting the culture. Second, that label reinforces the idea that the problem is intended for all students. Third, Frank and Anya suggest connections as a way to generate such problems — this would work in almost any situation, and does not rely so much on inspiration.]

In-class test problem-solving example:

f(x)=3(0.75)^x and g(x)=f(-x)+2

1. f(x) is plotted on the graph. Sketch g(x) on the same set of axes.
2. Write an exponential equation for g(x).

Some context for this problem: function transformations and function notation were part of a unit, about a month before exponentials. Students who were able to retain the information had fluency for both parts of the problem – exponentials and transformations. Putting them together is not difficult if you already know how to do it (our goal is not for this to be the most difficult problem on the test), but finding that connection on your own (the novelty of the problem) is a kind of problem solving.

Here are some more examples of problem solving from tests we have given in other classes.

Engaging with problem solving requires engaging with uncertainty, and that will never be easy for high schoolers. It’s not that students were not problem solving in math class, and now they are. Rather, our goals are to promote engagement and help make explicit to students that they are practicing hard thinking which will serve them in and out of math class.

[Yes, understanding and sharing these goals is essential if we are to generate buy-in for a sufficiently challenging and effective math program. Given societal pressures, a strategy for including problem solving throughout, including as part of assessment,  is just as important as creating or finding worthwhile problems for students to solve. Thanks, Frank and Anya!]

If you have any questions, or have tried something similar, reach out to us at asturm@ma.org and fcassano@ma.org!

These experiments rarely last more than a few years for various reasons: they fail in one way or another, they are difficult to staff, they do not match district or state standards, and so on. Quite often, they water down the disciplinary content by lowering expectations. For example, high school students get to review percents and proportional reasoning, or such topics as mean vs. median, when those are actually middle school topics.

A more successful approach is the teaching of parallel, disciplinary, courses which reinforce each other. An example might be a History course on the US in the twentieth century alongside an English course on American literature from that time period. Or an introduction to the physics and mathematics of motion offered in those two departments. But again, those experiments are usually short-lived, as they entail complications for the scheduling of both teachers and students.

While I respect the impulse behind those attempts and admire the ambition of those involved, I think that we serve our students better when we take a different approach: unwavering  disciplinary priorities within departmental offerings, combined with genuine attempts at making connections with other fields within those courses. It is not difficult to find math textbooks with grade-level content that make such connections. I got great ideas along these lines in the books by the University of Chicago School Mathematics Project, in the works of Paul Foerster, and in many other resources. Those ideas influenced my own curriculum development, as you can see for example in Algebra: Themes, Tools, Concepts,, and in my Algebra 2 and Precalculus materials.

Such connections are typically to physics or social studies, but there is no reason to stop there. I once co-taught a half-semester “Math and Art” high school elective, along with an art teacher. (As is to be expected, we only got to do this once or twice.) Here is the course description:

Both mathematics and art investigate pattern and the nature of space. In many cultures and eras the two have inspired each other. This class will study: the golden ratio, squaring the circle, regular solids (Ancient Greek); a sophisticated understanding of symmetry (Islam); Latin and magic squares (Middle Ages); projective geometry (Renaissance); group theory (nineteenth century); Escher, recreational math, computer graphics (twentieth century). Students will use the lessons as a starting point for creating their own art.

We did not get very deep into some of these topics, but I was able to develop some interesting high school-appropriate lessons within that framework. Once the course could no longer be offered, a nice payoff was that I was able to inject some of its content into other classes, as grade-level math-art connections:

• Basic geometry. Squaring the circle yielded two activities: Leonardo’s Areas and Squaring Pentominoes. (Unfortunately, only the first made it into my geometry class.)
• Algebra 2. Projective geometry yielded the Perspective lab activity.
• Advanced geometry. Regular solids, symmetry, and a basic introduction to groups became core topics in my Space course. (On my website, see also Symmetry and Abstract Algebra.)

Does this positive experience contradict my overall critique of interdisciplinary learning? Not at all: this course was intended to complement, not replace, the classes offered by the math and art departments. This was genuine “enrichment” for the students who got to take the class, and took nothing away from the essential disciplinary learning that they pursued in other classes. It is an example of one way to do this right.

Here is a painting by one of our students:

— Henri

Henri, in your experience, what are the pros and cons of using function diagrams with kids (in addition to the more standard Cartesian representation)? 

My BlueSky reply: “Cartesian graphs are a life tool. Function diagrams are a learning tool, so less crucial, though I have found a few activities at various levels irreplaceable.”

Function diagrams represent functions by connecting points on a (usually vertical) x-axis with points on a parallel y-axis. If you’re not familiar with them, see this page on my website and follow the links. (Warning: it’s a bit of a rabbit hole.) Here is a diagram for y = 2x – 3:

I’ll use this post to answer Bryan’s question.

First, the cons:

• As with everything else not already in the curriculum, it does take time, so you have to sort out what it will replace.
• It’s a hard sell to colleagues. Perhaps because it’s not standard fare, I found it difficult to convince some in my department of the benefits of this representation.

And now the pros:

• Working in a different representation helps us reach a broader range of students. Function diagrams are not more difficult than the standard representations, and for some ideas they are more straightforward, so they provide an additional way into the concepts to students who are struggling. At the same time, they are interesting to kids who already understand the threesome of table, formula, and graph. The usual objection to teaching something important in yet one more way is that “it will confuse the students”. I have not found that to be the case — quite the opposite.
• Understanding obtained this way complements and reinforces that which is obtained the traditional way, especially if connections are made between representations. 

Here are four “best of” function diagram activities / lessons which yield a great educational payoff with a minimal time cost — one per grade level. All four of these activities also work well in professional development workshops. In fact, even if teachers do not end up using them with students, that representation expands and deepens their own understanding.

1. Nine Function Diagrams is an excellent conversation starter, as it triggers discussions of many issues that come up in the transition from arithmetic to algebra. The worksheet only involves “one-step” functions, so the focus is really on operations and algebraic notation — two foundational ingredients in that transition. I used it near the start of Math 1, a course my department offered to ninth graders who had not taken algebra in middle school. 
2. Sixteen Function Diagrams takes it a step further: it involves identifying “two-step” linear functions, in other words the m and b in the usual y = mx + b. What’s great about it is that it forces a deeper understanding of those parameters as students figure out and share strategies to speed up the process of recognizing the functions. I used it to review linear functions at the start of Math 2. Because the representation is unfamiliar, it is a way to review these concepts in a non-rote way that is interesting to all students.
3. Name That Function! asks the students to identify standard Algebra 2 / Precalculus functions by watching animated function diagrams. It is quite entertaining, and again makes a great conversation starter.
4. Finally, function diagrams provide a rich environment to discuss rate of change in a way that complements slope. And they allow for a representation of the composition of functions that is much, much clearer than is possible on a Cartesian graph. Putting those two things together reveals why the derivative of the composite function is the product of the two derivatives — the chain rule. They also make the idea of inverse function much more visual. One way to start that conversation is in this worksheet, perhaps also using the images in #2 above for rate of change.

There is much more about these and other activities, a bibliography, and many function diagram applets on my website, starting here: Function Diagrams. And as always, I’m available to answer any questions.

One more thing: on BlueSky, Bryan suggests that this representation can be used to see translations, dilations, and reflections in a 1-D version. That is of course mathematically legitimate. Pedagogically, my experience is that it is not the best way to introduce geometric transformations, because the lack of shapes to transform makes it much less engaging and interesting to kids. However, it is an effective way to test understanding of those transformations in a later activity or assessment, narrowing the focus to 1D after introducing it in 2D. See also this kinesthetic activity, which starts in 1D and introduces rotation and the second dimension.

— Henri

Previous blog posts on function diagrams:  Learning from Teaching  |  Asilomar Report, Part 2

Function diagrams home page.

Once again, it was a smaller conference than pre-Covid, and  I saw fewer familiar faces among the attendees. Some of my favorite presenters were not there, and there was little emphasis on mathematical content, so I found the program as a whole rather disappointing.  Still, Pacific Grove is a great place to be, and it was nice to see ex-colleagues, meet people who find my website useful, and say hello to a fan who had already purchased my new book. 

Here are some notes from that day.

Grace Kelemanik

Grace is famous as the co-creator of classroom routines such as “contemplate then calculate”, which Robin Pemantle and I referenced in Chapter 3 of There Is No One Way to Teach Math. In this presentation, she introduced a not-unrelated idea: give students two or three minutes of think time before they tackle a problem, and make clear that this is different from starting to work on the problem. It’s a time to “ask yourself” various (teacher-supplied?) questions before you jump in. (For example, “what am I being asked?”, “what is the important information I was given?”, and so on.)

At the other end, once the problem has been solved, she has students report their solution in pairs. One kid stays at their desk, and talks through the solution, while the other is at the board silently pointing at the relevant images, equations, or whatever. Another student can then be asked to rephrase or clarify, and this time the teacher may do the pointing. She calls this: “Let’s see what you’re saying”. 

These are “Engagement Routines that Promote Student Agency” (the title of the session.) They do not take much time, but they do slow down the pace of the lesson which allows more students to get on board. Zeroing in on such classroom discussion micro-techniques is a way to highlight what many of us do intuitively, or have learned from each other in classroom visits or from mentoring. Naming the techniques, as Grace does, makes it that much easier to promote them. 

For some of the techniques I use to maximize participation in class discussion, see these blog posts: Project SEED | Every Minute Counts.

Frank Cassano and Anya Sturm

Frank and Anya shared excellent Algebra 2 activities. They start the course on Day 1 with interesting number-based problems that require no knowledge beyond basic algebra. This helps set the tone for the course: the message to students is that this course will involve thinking! (See my Day 1 manifesto here.) They shared an anchor activity on exponential functions, one that is not unlike my Rolling Dice, and ended the session with a nice set of problem-based introductory activities for a unit on quadratics.

The main point of their session was about “Integrating Argumentation and Problem-Solving” into any math course. The idea is to have explicitly labeled “argumentation” (proof) and “problem solving” sections in daily work and in assessments.  This work is evaluated with specific rubrics that distinguish those skills from content mastery. One benefit of this policy is that it highlights what makes math important even to students who have no interest in pursuing a STEM career.

This is a department-wide practice, and thus students accept this kind of work as a normal part of doing math. It follows that they do not object to writing in math class, or to being asked to solve problems without having been told ahead of time how to do it. I hope Frank, Anya, and their colleagues will share this approach broadly.

Kevin Dykema

Kevin is a past President of NCTM, a long-time 8th-grade teacher, and a promoter of “productive struggle”. In fact, he opened his talk by pointing out that struggle is indeed part of learning anything worth learning, whether math, swimming, or riding a bike. He made clear that for learning to happen, the teacher needs to be right there, supporting the student: a swimming instructor does not throw the kid into the pool and walk away! (I made that point in my last blog post, but this is a great way to put it!) The session’s topic was “Manipulatives in Middle School”, and he likened manipulatives to training wheels. They are one way to support the learner, but they are not the goal of instruction.

I could not find a seat in the overcrowded session, so I left early on, but not before seeing Kevin’s introduction to Algebra Tiles. It involved telling students “we’re not in Base Ten Blocks land any more!” I wish he knew about the Lab Gear, which is largely compatible with Base Ten Blocks, but so it goes.

Henri Picciotto

My session was titled “Practical Strategies to Reach the Full Range of Students”. I managed to talk for 80 minutes about teaching heterogeneous classes, with only a few interruptions for participants to discuss this or that. If you have attended my workshops and presentations in the past, you’ll know that this is not typical for me: I usually reserve the majority of the time for attendee participation. Still, it went well, I think. I had intended to point to specific chapters in There Is No One Way, but as it turns out the topic required mentions of ideas from just about all the chapters. If you’re interested in buying the book, you should make your move, as the publisher offers a 20% discount until January 31 if you use the code SMA24.

For more information on the book, see these blog posts: 1 | 2.

For more information on that session, including many links to free materials and all the slides, see my Talks page.