Teaching from the novice perspective

Here are 3 things you can teach a 4-year-old (they’re real parts of Montessori pre-K) :

1. How to tie shoelaces
2. How to polish leather shoes
3. How to sew with needle and thread.

If you were designing a curriculum for 4-year-olds and you wanted to make a unit containing two of these topics, which two would you group together? Your choice says a lot about your perspective.

If you frame the unit from a functional perspective, you’ll group skills together based on when we use them. Shoelace-tying and shoe-polishing both relate to shoes that are fancier than a 4-year-old typically wears, so you might group them together with the theme of “Getting dressed up!” and plan some kind of fancy footwear party as a culminating activity.

We group topics functionally all the time in Algebra 1, too. For example, we have three different tools for solving systems of linear equations: graphing, substitution, and elimination. Since these three tools have related functions — they help us solve similar types of problems — we group them together in a single unit.

The problem with grouping functionally is that topics that are closely-related according to when we use them may be only distantly-related in their rationale, or in the steps students must follow to implement them.

Let’s go back to the 4-year-olds for a minute. Tying shoelaces and sewing with needle and thread aren’t related functionally, because real-life situations in which we need to tie shoes don’t overlap much with situations requiring sewing. But they do involve very similar student actions: fine motor control of strings. If you create a unit based not on the expert perspective (“When will I use these skills after I’ve learned them?”) but instead on the novice perspective (“What am I focusing on while I’m learning these skills?”), then I think you’ll group shoelace-tying and sewing together.

I believe most Algebra 1 curricula ignore the novice perspective, leading to units that don’t feel to students like they have any internal connectedness. This is a huge factor in students’ sense that math is a meaningless collection of separate skills.

Today I’d like to share an Algebra 1 unit on expressions and equations that goes deeper than beginning-of-year units generally go, while also giving students a sense of connectedness by sticking to the novice perspective. For comparison, let’s first take a peek at the traditional back-to-school Algebra 1 unit.

The traditional way

Take a look at the hodgepodge of topics in the first standard in Virginia’s curriculum. If this doesn’t show you what a conceptual mess can be made by adopting the functional perspective, I don’t know what will. The skills needed to master these topics are almost completely unrelated to each other. Everything in black is a direct quotation from the standards, and notes in red are my clarifying notes.

• A.1.a: “Translate verbal quantitative situations into algebraic expressions and vice versa.”
  • This means taking a description like “The sum of 4 and the product of 5 and three less than a number” and turning it into 4 + 5(x-3).” Ugh. This is essentially a grammar lesson about parsing the math version of how people get to know each other at family reunions: “Okay, so your parents are Aunt Betty’s son’s daughter and the grandson of the sister of my grandfather. ” The hard part is just diagramming the sentences in your mind.
• A.1.b: “Model real-world situations with algebraic expressions in a variety of representations (concrete, pictorial, symbolic, verbal).”
  • This means comprehending a situation like “Ed has saved $40, and he saves another $10 per week” — and knowing how to translate it into an expression like 40+10w. It’s a conceptual topic.
• A.1.c: “Evaluate algebraic expressions for a given replacement set to include rational numbers.”
  • E.g., substitute x = -3/4 into 100 – 2x² . This is a drill topic that can be taught with or without comprehension.
• A.1.d: “Evaluate expressions that contain absolute value, square roots, and cube roots.”
  • E.g., substitute x = -5 into  -2|x – 3|. Another drill topic.

So we have a unit about (a) reading the weird grammar of math expressions written verbally, (b) comprehending real-life situations and representing them with math, (c) substituting numbers into basic expressions, and (d) substituting numbers into more complicated expressions. If you squint real hard, you can see how these fit into a coherent unit. Do you see it? Each bullet point uses the word “expressions.” That’s it. That’s the rationale!

Let’s consider that from a novice perspective. When students are substituting -3/4 into the expression 100 – 2x², are they thinking, “This is a problem about an expression,” or are they thinking, “Remember to put the -3/4 in parentheses?” Since they’re thinking about what they’re supposed to do, these topics feel like they have almost no coherent theme. You might argue that (b) does logically follow (a). That is, since Ed has already saved $40 and saves another $10/week, students might start thinking, “Aha! So Ed’s savings will be 40 more than the product of 10 and the number of weeks.” But of course no kid would think that way. It’s an expert perspective.

You might say that at least goals (b), (c) and (d) fit well together as different aspects of the order of operations. But this early in Algebra 1, any real-life situations we cover will be modeled by a very simple linear expression, like 40+10x. For expressions this simple, the order of operations thinking that students do is basically trivial. The difficulty lies elsewhere (in the fact that story problems are always hard for kids to interpret). In contrast, the expressions in goals (c) and (d) are advanced order of operations questions involving multiple operations inside and outside square roots, cube roots, absolute values, etc. There is very little overlap in student thinking between the topics.

Teaching from a functional perspective isn’t always wrong

The functional perspective emphasizes when we use skills, not how we apply them or why they make sense. Let’s go back to the example of the typical unit on Linear Systems. By grouping together the skills of solving by substitution, elimination, or graphing, the unit communicates that these methods perform the same function, even though the steps that students follow to apply each skill are totally different.

Linking concepts based on when we use them is important, but I’d argue it’s best for tying skills together after you’ve taught them.

Simply put: we should design students’ first foray into each concept around why it makes sense or how we work with it; once the concept has been mastered, we should connect it to other concepts based on when we apply them.

Is this the way your district teaches?

In Virginia, I think it’s much more common to see a straight functional perspective through the whole year.

One caveat: topics with similar meanings or procedures shouldn’t always be taught together. Sometimes you need to spread them out over several months. This is especially true when they rely on a tough idea or a tricky procedure — you may want to give the first topic time to mature in students’ minds before advancing to other levels of the same kind of thinking. Even here, though, we’re assiduously focusing on how the class feels for novices: are their brains ready to continue this topic, or is the introduction still cooking in their heads?

At all costs, we need to prevent classes from feeling like this:

Too often school is like a parade – a series of lessons – followed by a test to see how well the crowd paid attention to the parade.

We can do better. pic.twitter.com/hMXX2VD5O3

— David Coffey (@delta_dc) July 23, 2018

How I planned for connectedness

If you’re still reading, it’s time for the goodies: an Algebra 1 unit that cuts across the traditionally-defined units in order to create a more coherent feel. Below I’ve circled the topics included. They’re shown against the background of the pacing guide of Henrico County, VA, to highlight how the unit cuts across traditional topics. (I don’t work in Henrico, but I do appreciate how much stuff they post online).

Here’s a quick overview of the unit:

• The first half of the unit flows from the order of operations without grouping symbols, emphasizing the role of exponents.
  • -5² versus (-5)². Although most students won’t learn about using the ± sign with quadratic equations until 4th Quarter, advanced students will be able to learn it at the end of this unit. So the beginning of the unit prepares them by showing that there are two different solutions to (…)²=25. Meanwhile, learning the difference between -5² and (-5)² gives everyone the conceptual basis for the next topic, below.
  •  Substituting negative or fractional values into expressions. Because of what we learned in the previous topic, you have to substitute negative values in parentheses, right? Here we practice that skill and even practice subbing possible solutions into quadratic equations to check which ones are correct.
  • Adding and subtracting polynomials. As you’ll see below, this topic emerges naturally from the previous one. We use Desmos-based algebra tiles. The Desmos ones let us focus on the main idea more easily than physical tiles do. Activity links are below.
  • Regular combining like terms problems, with distribution. This is just an extension of adding and subtracting polynomials.
• The second half of the unit flows from the order of operations with grouping symbols (except solving-by-balancing, which provides a brief, intentional pause in the conceptual progression).
  • Multiplying binomials. This is also taught with Desmos algebra tiles.
  • Building expressions with the order of operations. E.g., what expression represents starting with x, adding 3, squaring that result, and then dividing by 5? This topic includes literal expressions.
  • Solving equations by balancing. This also uses algebra tiles, though this time they’re physical ones. (By now, students understand them well enough that they’re quite helpful). Your students can draw pictures if you don’t want to use physical tiles.
  • Solving literal and nonlinear equations by unwinding. Now we reverse the order of operations to isolate x. This includes solving quadratics by taking square roots, but also solving easy radical equations and solving literal equations.

The unifying theme

The order of operations binds this unit together — it’s the basis for how we evaluate expressions, combine like terms, and solve equations. I don’t think most curricula highlight that connectedness, though. And I don’t think students in most classes realize that all of these topics emerge from a single, simple rule.

One particular aspect of the order of operations serves as a recurring motif for the unit. It’s this:

In an expression like 3x², the order of operations says the (…)² comes before the •3. 

Hopefully, the unit feels like a spiderweb made of many threads but arranged into a coherent whole. If so, expressions like 3x² are the stickiest thread. Early in Algebra 1, students’ experiences with x in math are almost entirely with linear terms. When they see something like 3x², their eyes gravitate toward the familiar part, 3x, and treat it as a unit. Their eyes perceive the (…)² as secondary, and that means they often interpret 3x² completely wrong. This unit is designed to give students a chance to get caught on this sticky thread over and over until their brains are ready to see 3x² for what it actually means.

Quadratics in 1st Quarter?

It’s very unusual to include polynomials and quadratic equations in the 1st quarter of Algebra 1. What could possibly justify doing that? Well, quadratics is the hardest unit for most students. It includes these 5 topics:

1. Factoring trinomials
2. Advanced factoring (e.g., factoring out a GCF first)
3. Solving quadratics:
   • by factoring
   • by taking square roots (and using the ± sign to get both solutions)
   • with the quadratic formula
4. Knowing which solving method to apply when
5. Parabolic graphs

From the novice perspective, grouping these topics together is a disaster because the skills related to factoring feel very disconnected from the other skills. This disconnect is hugely meaningful. It’s an idea that undergirds not just this unit but also the rest of high school math: when an equation includes terms of different degrees (like x² and x), then there’s no way to isolate x, so we have to solve by applying the zero-product property. How foundational is this? Well, the phrase “Fundamental Theorem of Algebra” comes to mind, right? We cannot let students finish Algebra 1 unclear on why suddenly in April, when we have just come back from spring break and are starting to think of long summer days of video games and Netflix, we randomly decided to invent a crazy-complicated technique like solving by factoring. We cannot let students decide that it’ll be okay just to ride out the year and get those questions wrong on the final exam. And we don’t want students to be overwhelmed by the difference between solving by factoring and the other methods…we want that difference to provoke a fundamental shift in thinking.

So I’m moving solving by square roots all the way up to 1st quarter while saving factoring and the quadratic formula for 4th quarter. That way when students see 1+(x+4)²=0 and x²+5x+4=0 side-by-side in 4th quarter, they’ll recognize that the first equation is just another example of solving-by-undoing, but the second equation is radically different and will require a new technique. Unless the first equation looks familiar and the second looks weird and somehow wrong, that recognition won’t occur. It’s also really good to teach multiplying binomials early in the year, because then the skill has automaticity by the time students attempt to reverse it when they learn factoring.

In first quarter, I’ll teach just how to get one solution to 1+(x+4)²=145, without using the ± sign. Then in 4th quarter, we’ll teach the ± sign part and connect it to the two zeros of a parabola. This approach doesn’t just pay dividends in 4th quarter. It also makes 1st quarter more meaningful and interesting from a novice perspective, because solving by taking square roots is so similar to solving literal equations. Both are aspects of the order of operations — they belong together. And only when students form connections like this can they experience math as making them powerful.

There and back again: a journey through the order of operations

Here’s a detailed look at the unit, with links to key materials:

1. Topic 1:  -5² versus (-5)².
   • Why does -5² have a negative value, while (-5)² has a positive value? Because (-5)² means (-5)(-5), but -5² means -1•5². And in expressions like -1•5², you do the exponent first, so there’s only one negative sign, not two. This is the sticky spider web thread highlighted above.
   • An even number of negative signs produces a positive answer, but an odd number of negatives makes a negative answer. So would the following values be positive or negative:  -5¹²⁰ ,  (-5)¹²⁰ , and (-5)¹²¹ ?
   • What about questions implicitly relying on exponent rules, like (-5)³•(-5)¹⁵ ?  Even without having been taught exponent properties, students can see that this will have 18 negative signs, so it will be positive because 18 divides evenly into pairs. Our spiderweb has caught some prey: the intuition of exponent rules! We’ll eat this prey in Topic 4’s extension.
2. Topic 2: You must substitute negative values and fractions in parentheses. 
   • To evaluate x² + 5x for -3, you need parentheses: (-3)²+5(-3). That’s because (-3)² is not the same as -3², as we learned in Topic 1.
   • To evaluate it for 2/3, you also need parentheses: (2/3)²+5(2/3), for the same reason.
   •  You can check solutions to quadratic equations by substituting them correctly. For example, to check whether -1 is a solution to x²+5x+4, you can evaluate (-1)²+5(-1)+4. Remember to use parentheses.
   • Extension for advanced students: some students can now begin trying to guess both solutions to equations like 2+(x+4)²=27. Guessing the answer x=1 is easy, but guessing the answer -9 takes thinking. This lays the groundwork for an extension in Topic 9.
     • Commentary interlude: almost all Algebra 1 curricula put Topic 2 in the back-to-school unit, which is generally presented more broadly to include other functions such as absolute values or square roots and cube roots. What’s unique above is that I haven’t included those other functions (yet) and instead included Topic 1 to draw out our sticky thread, in part for comprehension of Topic 2 and in part because then everything connects so well to…  
3. Topic 3: Adding polynomials. Even students who’ve learned how to add polynomials often struggle to remember whether x²+x² equals 2x² or 2x⁴ . Adding polynomials is normally taught at the end of the year (see the Henrico curriculum), but we have a golden opportunity for connecting to Topics 1 & 2. Our sticky thread says that in the expression 2x² , squaring the x comes first and the 2 just doubles that result. So 2x² literally means x²+x². Here’s how we use Desmos to show it visually:
   • First, we combine like terms with the numerical expression 4²+4+4+4² (see video below): 
   • Next, we generalize to expressions involving x. Check out the visualization of x being a value that changes!
   • Here’s a link to the Desmos materials.
   • I like UW-Madison professor Martha Alibali’s explanation of the power of algebra tiles: “Algebra tiles can highlight mathematical relationships that are often difficult to ‘see’ in symbolic representations…[P]hysical representations can make aspects of mathematical situations salient in ways that foster sense making and support learning.”
4. Topic 4: The distributive property and combining like terms.
   • This includes subtracting polynomials, plus all other combining like terms and distributive property problem types. Subtracting is treated as distributing a -1 and then adding.
   • I like to include combining like terms questions that incorporate the order of operations, like 3 + 2•5x + 8 (remember to multiply before adding). Students need to see combining like terms as just an aspect of the order of operations.
   • Extension for advanced students: Advanced students who have mastered this can begin working on exponent rules drills, which were teased in Topic 1. Students who do this must be able to add polynomials perfectly, so they know beyond a shadow of a doubt that x²+x²=2x² .  If they’re 100% solid on this, they may begin working on facts like x²•x²=x⁴ . Other students will learn exponent rules later in the year, but students who start learning them now will advance to more difficult problems.
5. Topic 5: Multiplying binomials. This is also usually taught at the end of the year, but since we’ve been discussing how to visualize x•x as x² , it’s natural to visualize expressions like x(x+1) , too. The segue to multiplying binomials feels smooth and connected to students.
   • Here’s a quick preview of the lesson:
   • Extension: To really unify what’s been taught so far, you can ask students to represent the product 3(x+1)² either with algebra tiles or as an expression. The 3 just tells you how many copies of (x+1)(x+1) to make. The sticky thread again!
   • Here’s the lesson on Desmos.
6. Topic 6: Evaluating Absolute Value and Radical Expressions. This covers the topics we skipped back in Topic 2. They take us away from the sticky thread and turn our attention toward the rest of the order of operations.
7. Topic 7: Building expressions with the order of operations
   • In this necessary twist on the typical order of operations unit, students learn to read algebraic expressions not left-to-right, but starting from the position of the x. For example, given the expression 4+3x² , students should say it starts with the x (not the 4), then squares it, then multiplies by 3, and finally adds 4. Even students who can correctly compute 4+3(5)² numerically will often misinterpret 4+3x² when it’s presented algebraically, so this takes practice.
   • Topic 7 is a critical prerequisite to teaching students to solve equations by “unwinding” them, because unwinding expressions means applying the inverse operations in the reverse order. First they must learn to read expressions in forward order.
   • Here are sample questions for this topic:
   • Please notice the last question in the pic above, #4. Literal expressions! While most curricula introduce literal equations only to solve them, here we spend several weeks just building literal expressions in the forward order. Only when you can build them are you ready to unwind them to isolate a specific variable.
   • Extensions for advanced students. To model a real-life situation with an algebraic expression, you’ve got to know the order of computational steps required by the situation. So this is the point where you could include modeling questions like the story problem discussed above, with Ed saving $10 per week on top of $40 already saved. Personally, I’d save this particular scenario for the linear functions unit since it has a clear slope and y-intercept. But I’d absolutely give other scenarios to students who are ready for an extension after mastering Topic 7. Here’s a Desmos extension lesson, Mystery Area, that I made for just this. After Mystery Area, I’d let students try Desmos’ Central Park. Mystery Area should be done by most students (anyone who’s not busy remediating other weaknesses during this extension phase of the unit), while Central Park should be attempted only by those who feel they’re ready for a challenge. Here’s a third extension that lets students think about how to interpret something like 3(x+1)². I’m still editing this lesson, but you’re welcome to check it out. It’s for advanced students only, I think, and I think it needs some revision.
8. Topic 8: Solving equations by balancing. This topic is intentionally placed out of order to pause the conceptual progression. We have just finished a topic on building expressions. In the following topic, we’ll learn to unwind them. Since unwinding expressions requires reversing the order used to build them, we need to give students a little extra time to hone their building skills. Inserting solving-by-balancing at this point provides the necessary delay to allow that skill to mature. The balancing steps we follow here will be modeled with algebra tiles, so the topic will feel natural with the flow of the unit.
   • Lots of teachers hate using algebra tiles for solving equations, and I get why. Kids play with them (which is why all the lessons up till now have used Desmos virtual tiles). And the first couple times kids use them, tiles can feel more confusing than written algebra because you have to count out all these little pieces and move them around. That’s why I never let kids touch the tiles till we’ve practiced the moves by drawing pictures. Animated powerpoints are invaluable here because I can click through the steps while I walk around and help kids follow along. Here’s a short sample of my intro powerpoint:
   • The reason to use algebra tiles is to help students understand the role of the = sign in equations. Say a kid’s trying to solve 1+4x+3=8. Here are a few mistakes all involving the = sign:
     • Combining 1, 3, and 8 across the = sign.
     • Subtracting 1 from the left side twice (below the 1 and the 3)
     • Subtracting 8 from both sides (below the 3 and the 8) and, instead of writing 0 on the right side, just moving the = sign to say 1+4x= -5.
   • When students make these mistakes, it’s an indication that they don’t understand how the = sign signals balance between the sides of an equation. Which means they may not have fully internalized what equation-solving is actually about. Algebra tiles allow students to develop this understanding progressively by situating the problem within a context of balancing. Within their first lesson using tiles, student go from solving equations without like terms to solving ones like 2x+15=x+3+3x, and the tiles allow them understand what you can and can’t do with each term.
   • Links: Here are lesson materials for the intro lesson (.pptx, student handout). After the intro lesson, we consolidate what we’ve learned about respecting the role of the equals sign with these materials (.docx and .pdf). Finally, we  transition from algebra tiles to symbols (.pptx, student handout, teacher notes).
9. Topic 9: Solving equations by unwinding them. Instead of using a balancing metaphor here, we’ll isolate x with an unwinding metaphor: getting x by itself by using the inverse operations in the reverse order. This allows us to tackle nonlinear equations, including quadratics that can be solved by taking square roots, but also including all sorts of other forms: equations with nested fractions (see the picture below), or radical signs, or even quadratics/radicals within nested fractions.
   • Intro: you have to unwind in the reverse order. I have a fun intro lesson to Topic 9 based around number tricks that ask you to pick any number, follow a bunch of computational steps, and then give you back your original number  (.pptx, handout .docx, teacher .docx, handout pdf, teacher pdf).
   • Main lesson: Now students learn to solve by unwinding. They’ll solve all different kinds: radical equations, equations with complex fractions, and quadratic equations. This is a flipped lesson — students watch a video I made and fill out the notes. Materials here (handout .docx, handout pdf, lesson video, answer key pdf, answer key .docx). Here’s a sample screenshot of the process students learn to use:
   • Literal equations: Literal equations flow directly from this. If you want to solve 4(a + m)²=C for the variable a, you just list the steps to build the left-hand side starting from a, and then you do all the inverse operations in the reverse order. It’s the same concept as solving by unwinding. Since most curricula, e.g., Henrico, teach literal equations in the 1st quarter, they also ought to teach other unwinding strategies in 1st quarter…notably, solving quadratics by taking square roots. As I said earlier, I don’t plan to teach using the ± sign to find both roots of a quadratic here. We’ll just practice finding the easier root for now. In 4th quarter, we’ll introduce the ± sign and connect the two roots to the intercepts of a parabola.
   • Extension for advanced students: Because Topics 1 & 2 focused on the difference between -5² and (-5)² , advanced students can learn about using the ± sign if they’re ready. This video is my flipped lesson on the topic, so it can be mostly self-guided for advanced students working together. Meanwhile, other students will work with me to remediate their weak spots.  When we cover the ± sign in 4th quarter, students who have already done it will do a different enrichment activity.

By the end of the unit, I want students to be like a spider, able to dance across this web without ever getting stuck. Able to repair any connection that breaks by spinning new thread. And able to sit in the center calmly regarding the whole through a multitude of eyes.

Classical music connection

Have you ever listened to Antonín Dvořák’s 9th symphony? Each melody is striking as you hear it. But Dvořák then weaves these themes into a coherent whole with such depth that it’s virtually impossible to perceive all the connections at same time, giving us this sense of something complete, something absolute, which we can comprehend beautifully from any angle, but the whole of which we can never see.

You can get a taste, and some explanation of the interwoven themes, below. I wish math class felt like this:

Connections to other teachers’ ideas

Jonathan Claydon is also busy reordering topics to show math’s internal connectedness. Here’s his summary of the talk he gave at NCTM ’18 on his work.

In contrast, folks at Illustrative Mathematics (hi, Tina!) frequently press the point that busy teachers can almost never knit a curriculum together successfully — that teachers really should focus on implementation and adaptation. Here’s a blog post in which Brooke Powers gave her experience of letting go of the curriculum reins and following the IM curriculum.

Rose Roberts wrote recently about how she’d like to group Algebra 1 topics into themes.