The Pointlessness of Precalculus

I know! Algebra is not used every day. People mock its pointlessness all the time. This showed up on Facebook the other day. You can spend your life escaping algebra! Woo hoo!

What we learned in history, science, and poetry is not useful either. When do you use a historical date? Recite Poe? Balance chemical equations? Calculate acceleration? But, I digress.

I was tutoring a senior (Jay) in precalculus while a junior in the same class (Kay) watched. The following conversation ensued.

Jay: "Mrs. Tammy, when am I ever going to use this?"

Me: "I used this sort of thing to learn high level statistics, and my husband did to learn engineering."

Jay: "I'm not going to major in any of that. This is useless!"

Kay: "My mom teaches math. She never uses this stuff."

So, I admitted what everyone knows. We'll never need to prove trigonometric equations in real life. I shared what I love about these problems. It's like the Gordian's knot. Tug at the right spot, and the whole thing unravels quickly. The same thing that drives Sherlock Holmes to solve mysteries drives me to prove equations. A puzzle. A mystery. A quest. How exciting when you find the right key (or two) and the knots disappear!

Jay had to prove the sides of the equation to be equal even though they look different. Many manipulations shows they are the same. This looks much scarier than it really is!

I ask students what to do first. If I hear a confident "cross multiply," we continue. If I hear a faltering "Cross multiply?" doubt arises. Some students speak jargon that confuses them. So, they launch a buzz word, hoping it's the right one. Then, I ask why cross multiplication. If the explanation makes sense, we move ahead. Otherwise, we linger.

I ask what they can do to make the equation easier. "I don't like fractions!" [That is why we cross multiply.} Then, I ask how to dump the fraction. Most know to multiply the fraction by its denominator (the bottom). I probe further to see if they know why you multiply both sides by the denominator.

The reason so many students falter in math is that they do not understand the why behind what they are doing!

Imagine the left side of the equation sits on the left side of a balance, and the right side on the other. Multiplying the left by a number greater than 1 increases its weight. Doing the same thing to each side keeps the balance. This applet shows this for simple equations! Some pre-algebra materials sow the idea of seeing equations on a physical scale and viewing them as puzzles.

The joy of math is found in unraveling complex mysteries.

Multiplying both sides of the equation by sec θ - 1 scares some students. Working with fractions, however, is scarier! We treat this ugly expression the same as any simple number. Including parenthesis avoids confusion.



We multiply by sec θ - 1 to create sec θ - 1 over sec θ - 1.



Why? Well, what is sec θ - 1 divided by sec θ - 1? Relax! Take deep breaths. What is 2 ÷ 2? One! What is 300 ÷ 300? One! What is a million ÷ a million? One! What is cheese ball ÷ cheese ball? One! So, what is (sec θ - 1) ÷ (sec θ - 1)?

ONE!

See that's not so hard even if the fraction looked ugly. Now, we have rid ourselves of fractions and all is well.



Some eye the right side of the equation nervously. Why would anyone want to times one number by the difference between two other numbers? Suppose you find your favorite microwave lunch on sale for $2.94 and you want to buy one for each day of the week. You have $20.50 cash on hand. Do you have enough money? While Pamela might be able to multiply 2.94 by 7 in her head, lesser minds like mine can alter the problem slightly and solve mentally, too. The product of 2.94 and 7 is really seven rows of 2.94 as illustrated below.



If we alter 2.94 a tad, we have something easier to manage mentally. What? How can 3.00 - 0.06 be easier than 2.94?



Mulitiplying 3.00 - 0.06 times 7 looks like this.



We can multiply in our head! Yes, 7 times 3 dollars is 21 dollars and 7 times 6 cents is 42 cents. Taking 42 cents away from 21 dollars yields $20.58. Alas, you can only buy six meals.



On the left is another way to illustrate the distributive property. This little rabbit trail has a point for those who do not like the look of cot Θ (sec Θ - 1). You can try to multiply vertically if horizontally worries you.



Sadly, the equation is still ugly. Do you see the next step? Take one step, and the equation looks prettier. Try to focus on what you can remove completely. Think about the analogy of the balance and what you can take off both sides.



Each side of the equation has - cot Θ. Are you stuck? Pretend it's something less scary like - 2. What can you do to - 2 to turn it into zero? Add two! What is - 2 + 2? Zero! What is - 300 + 300? Zero! What is negative million + positive million? Zero! What is negative cheese ball + positive cheese ball? Zero! So, what is negative cot θ + positive cot θ?

ZERO, zip, nada!



The new form of the equation looks almost friendly compared to the original rubbish. At this point in the game (to me, it is a game), I convert everything on the right side to sin θ or cos θ and let the chips fall where they may. These little identities are things that you memorize with use and can always look up if you aren't suffering through a test: csc θ = 1/sin θ and cot θ = cos θ/sin θ and sec θ = 1/cos θ. I simply plug and chug on the right side from here on out!

Helping Others Find Joy in Math

Last spring, a friend of mine from church came to me a few weeks before finals for help in high school mathematics. Jay is a bright student who has earned passing grades in math by memorizing formulas without understanding and cramming for tests. I usually prefer to peel back the layers of knowledge to what a student understands and build up from there. With so little time, I had to focus on major habits and enough understanding to pass. I prioritized teaching what would give her the most bang for the buck in the little time she had before finals. She passed but we both knew she could have done better.

Even though she is not taking precalculus until next semester, Jay came to me for help this week. She has set a goal of earning a 90 or above in all her classes in her last semester of high school. We both know she has gaps in understanding, relies too much on the calculator, and mixes up formulas. Rather than revisiting difficult topics while under the pressure of mastering new material, she is going to build a solid base now when she can relax and learn.

I asked Jay what concept she would like to study first. "Negative numbers! I know what to do when you multiply and divide. I get confused about what happens when you add a negative and a positive. I never know what the sign should be."

I drew a number line, and we did some simple problems. I said, "Explain to me what happens when you add two positive numbers like +5 and +3." I illustrated the problem on the number line.



She told me, "When you add two positives, you head in a positive direction. They stay positive."

We did the same thing for adding two negative numbers like -5 and -3. Jay gave a similar explanation of heading in a negative direction and staying positive.



Then, we worked on the piece of the puzzle that had long mystified Jay. I plotted adding +5 and -3. Jay looked at it for a moment and said, "Oh, I see. The negative number is not large enough to cross zero. So, the answer is positive."



We then studied what would yield a negative answer: adding +3 and -5. Jay smiled, "I get it! Since 5 is larger than 3, it is going to cross zero and the answer is negative! Wow!!"



We continued pursuing this line thought by adding -5 and +3 and then -3 and +5. I gave her a couple of problems to make sure she could apply what she understood.





I like to give students other ways to understand a problem. Jacob's Algebra offered an alternative view of adding positive and negative numbers. The book depicts a number as a set of circles. A positive number has circles with no filling (white), and a negative number has filled-in circles. The picture below shows two numbers: +6 is the first row of circles and -6 is the second row. I drew a picture like this for Jay, and she saw immediately how adding them together results in an answer of 0.



I drew another picture with -2 in the first row of circles and +2 in the second row. Again, she saw immediately how adding them together results in an answer of 0.



The picture below illustrates -5 plus +16. Jay had no trouble explaining that the answer had to be positive since the number of positive circles is greater than the number of negative circles. Since there are 11 circles left, the answer must be 11.



The picture below illustrates +3 plus -8. Jay easily explained that the answer had to be negative since the number of negative circles is greater than the number of positive circles. Since there are 5 circles left, the answer must be -5.



I assigned several addition problems for her to solve. Jay got them all correct. Even better, she smiled the whole time because she didn't have to guess the sign of the resulting sum. Then I picked a more difficult problem from the book:

1 + -3 + 5 + -7 + 9 + -11

She drew a picture like the one below and said the answer had to be -6.



Since Jay needs practice with mental math, I processed the problem aloud in a different way for her, "Do you see that 1 plus 5 plus 9 is 6 plus 9, and that is 15?" Then, I wrote down the number 15. I said, "And, -3 plus -7 plus -11 is -10 plus -11 and that is 21." I wrote "+ -21 = -6" to finish the solution. To practice both ways of adding positive and negative numbers, I assigned the next problem in the book.

-1 + 3 + -5 + 7 + -9 + 11

She drew the following picture and told me the answer was 6. Then, she did some mental math and wrote down the equation, "21 + -15 = 6."



Then, she paused and said, "That's funny. You get the same answer only the signs are reversed. Oh, wait! All the signs are opposite. Had I thought about it, I wouldn't have had to do all that work. That's cool." She warmed my heart when she said, "I want to keep this paper! It finally makes sense!"

Before closing, I will summarize some of the important ideas that help people who may not care for math to experience joy.

• Figure out what the student knows. I usually do this by asking them to explain things to me. When they get, I ask a question about a more foundational concept until I find the student on solid ground.
• Take small steps of logic by demonstrating the process with objects or in pictures. Too many students think in memorized procedures and formulas but cannot explain the rationale behind them. Once they understand why, procedures and formulas become effortless.
• Ask the student to explain the rationale behind what they are doing. If they cannot explain it, then they lack a solid foundation.
• Show how to solve a problem in different ways. It helps students see how math can be a creative process.
• If a student struggles with mental math, assign problems with simple numbers and avoid using the calculator. I find that students who rely on calculators for simple problems usually lack number sense.

Breathing the Rarified Air of a Mountainous Land

David is a student at Charleston Southern University, taking 16 credits in your average freshman schedule. The main reason why he dipped back into math last spring was that his ACT math scores were less than ideal. I will refrain from sharing the actual number, but the folks at CSU felt he needed remedial math. I agreed. I suggested he retake the ACT to boost his math score and avoid a class in which he would study for long hours but not receive any credit. He pointed out that his college of choice had already accepted him. I did some digging and let him know that CSU offered a math placement exam. If he passed the test, he could take a regular math class. He saw wisdom in that idea and spent the spring and summer diligently studying algebra.

Why would the son of an engineer (my husband has three engineering degrees) and a mathematician (I have two math degrees) fare so poorly in math? For a long time, I have suspected that, while David has the intelligence to do well in STEM subjects, he prefers more creative outlets. He tackled math by memorizing and cramming, even though we had always encouraged him to think. Since he always seemed to struggle with math, I could never prove my hypothesis.

Long story short, David studied Jacob's Algebra from cover to cover. He was surprised to find the book, both interesting and readable. David told me that Jacobs teaches you how to look at problems logically and to understand algebra. For years, he focused on cramming formulas, which he mixed-up on comprehensive tests. Rather than feeling anxious when he couldn't think of a formula, he learned to relax, write down ideas, and reason his way through problems. He also realized that asking for help is not a sign of weakness and reached out rather than stare at a page until his brain froze.

I think Jacob's Algebra is a fabulous book for many reasons. The author has not updated the book since the late 1970s. Unlike textbooks of today, it does not have ADHD, distracting students with irrelevant graphics, sidebars, etc. When I peeked at David's college math text, the glossy pictures of movie scenes in which mathematics played a role nauseated me. The money wasted on pop culture would be better spent showing students how to reason. I personally think all the eye candy in the world will not attract students to this much maligned subject. Force-feeding formulas to children without understanding and encouraging them to cram to get ready for standardized tests make them loathe math. Jacobs carefully unfolds algebraic thinking through brain teasers and puzzles before getting abstract. Learning to reason and apply logic changed David's attitude toward math.

He also struggled with fluency and accuracy. The month before taking the placement test, I wrote algebra quizzes based upon ACT problems. I had him run through problems at Khan Academy to practice fluency and accuracy. Because he understood more, he had to memorize less. He could focus his attention on practicing and improving these habits.

The end result is that he earned an 81 on the placement test and is taking freshman math. But, even more important, he enjoys mathematics now. When asked what about his favorite classes, he put English Composition right at the top followed by math: "Actually for some reason I'm preferring the math problems over reading... Pretty odd, huh?" He dislikes reading because he is on a steady diet of typical textbooks (yawn). He even added, "I might take some extra math classes as soon as I get all my required math courses out of the way."

This fall, I have another opportunity to win another student over to mathematics. I will share the beginning of her story in my next post.

Another realm open to Intellect has an uninviting name, and travelling therein is difficult, what with steep faces of rock to climb and deep ravines to cross. The Principality of Mathematics is a mountainous land, but the air is very fine and health-giving, though some people find it too rare for their breathing. It differs from most mountainous countries in this, that you cannot lose your way, and that every step taken is on firm ground. People who seek their work or play in this principality find themselves braced by effort and satisfied with truth. ~ Charlotte Mason

Purple Math Age Problem

I've been assessing Pamela's word problem solving abilities and I'm pleased with what she can do with simple one step problems requiring whole-part thinking. In the process of assessing her, I've been transitioning her to the Singapore Math bar model concept that I explained on Wednesday. First, I assessed if she recognized whole and parts in ordinary situations that she understands. Pamela loves making her own lemonade, so this set-up was easy for her to do. I gave her this:

She easily set-up a whole-part model.

I gave her several problems like this and she set every single one up correctly. Then I gave her addition (unknown whole) and subtraction (unknown part) word problems based upon bar models she set up. Here is the one for lemonade problem:

Lemon juice has 11 calories. Lemonade has 165 calories. Sugar has 154 calories. How many calories does water have?

She correctly recorded the numbers in the model and figured out that water has 0 calories.

Once I was satisfied Pamela understood these problems, I gave her problems with distractors such as the following:

Mom baked the crust for 10 minutes and the filling for 55 minutes. She read a book for 15 minutes. How long did it bake?

She spotted the distractors right away and began crossing them out. Not all problems had them. I think from now on, I will include distractors occasionally so that she realizes she doesn't have to use every piece of information in a problem. Now, I plan to assess how well she applies the bar model graph to these concepts. I have a feeling we will be charting new territory in some cases: comparison, change, remainder, equal, excess value, repeated value, constant difference, constant quantity, and constant total.

The reason why I am so excited about helping Pamela learn to think in pictorial models is that they can solve many word problems covered in algebra without using letters and numbers. Once she understands them pictorially, I suspect the transition to letters and numbers will be easier for her. Here is an age problem I found at Purple Math. Notice that Purple Math sets up a system of equations and solves it. Now compare their method to a pictorial method, that is not quite Singapore Math and not quite Jacob's Elementary Algebra.

In January of the year 2000, I was one more than 11 times as old as my son William. In January of 2009, I was 7 more than 3 times as old as him. How old was my son in January of 2000

We will represent William's age in 2000 with an empty box:

2009 is 9 years later, and you can represent his age in this way:

In 2000, his mother was 1 more than 11 times his age, which is 11 empty boxes. You would add 1 to that to represent her age:

To represent her age 9 years later, increase the 1 to 10:

This table organizes all the information except for the last relationship we will analyze next:

In 2009, his mother was 7 more than 3 times as old as his age in that same year. So, you have to triple that age by writing the empty box and 9 3 times and add 7 to that:

We also know her age in 2009 was 11 empty boxes and 10 from the table we set up. All we need to do now is rearrange boxes and numbers until it works:

Rearrange the first line to match up the elements better. Compose 9, 9, 9, and 7 to make 34 and decompose that to 10 and 24.

Separate 24 into 8 equal parts yields 3.

That means an empty box is the same thing as 3. William was 3 in 2000. His mother was 34. In 2009, he was 12 and she was 43. Her age in 2009 (43) is 7 more than 3 times his age, or 7 + 3x12.

Now, here is the challenge for anyone wishing to try. Can you solve the other next age problem at Purple Math through pictures? If you email me your work, I will include it in my next math post.

In three more years, Miguel's grandfather will be six times as old as Miguel was last year. When Miguel's present age is added to his grandfather's present age, the total is 68. How old is each one now?

Pictorial Ways of Solving Algebraic Problems

A farmer had twice as many ducks as chickens. After the farmer sold 413 ducks and 19 chickens died, he has half as many ducks as chickens. How many ducks does he have now?

I have been studying ways to solve a simple problem in algebra pictorially rather than the traditional methods as I think through how to teach Pamela how to do this one day. Since I know many folks scared off by high school math occasionally pop into my blog, I was wondering if this makes more sense to you than the traditional method which I list at the end of this post. This pictorial method incorporates two strategies: (1) working backward from the answer and (2) incorporating the techniques taught in the first four chapters of Jacob's Elementary Algebra. Pamela has been using an empty box for the unknown for years, based on how Making Math Meaningful teaches whole-part thinking.

Ducks
We want to know the number of ducks the farmer has now, which is our unknown, represented by the empty box.

We know that the farmer used to have 413 more ducks than he has now because he sold that many. So, at the beginning of the problem, the farmer had whatever number he has now and the 413 he sold.
 

Chickens
We want to know the number of chickens the farmer has now. We know that he has half as many ducks as he does chickens. If he has 8 ducks, he would have 16 chickens. That means that the number of chickens is double the number of ducks. Whatever the number of ducks is now, the number of chickens is twice the amount.

We know that the farmer used to have 19 more chickens than he has now because that many chickens died. So, at the beginning of the problem, the farmer had whatever number of chickens he has now and the 19 that died.


Summarize the Quantity of Ducks and Chickens Before and After

We were given one more relationship: at the beginning of the problem, he had twice as many ducks as chickens. If he had 8 ducks, then he had 4 chickens. That means the number of ducks at the beginning was twice the number of chickens. Twice the number of chickens at the beginning would be double of what is in the table, or


We can rewrite twice the number of chickens as,


The number of ducks at the beginning of the problem is the same as double the number of chickens,


We can decompose 413 in a way to help us see the answer:
413 = 375 + 38 = 125 + 125 + 125 + 38



That means 125 goes into the empty box, which is the number of ducks that the farmer has now.

To check our work, we can plug 125 into the empty boxes into the table. The number of chickens now is twice what was in the empty box, 2 x 125 or 250. The number of ducks at the beginning was the empty box and 413, 125 + 413 or 538. The number of chickens at the beginning was the twice the empty box and 19, or 2x125 + 19 or 269.

Does this make sense? At the beginning of the problem, the number of ducks (538) is twice the number of chickens (269). At the end of the problem, the number of ducks (125) is half the number of chickens (25). All relationships make sense!

Other Methods:
I found this problem at a Singapore Math blog, which offers multiple problem solving strategies that are more pictorial than the traditional method. The author linked to a more thorough explanation of Singapore's model method that you might enjoy.

Traditional Method:
Let c be the number of chickens and d be the number of ducks at the beginning of the problem. Since there are twice as many ducks as there are chickens at the beginning,
d = 2c

Since the farmer sold 413 ducks, the number of ducks at the end of the problem is the expression,
d - 413

Since 19 chickens died, the number of chickens at the end of the problem is the expression,
c - 19

We also know that now there are half as many ducks as chickens, so we can write an equation for this relationship:
d - 413 = ½(c - 19)

Substituting the first relationship between ducks and chicks, we can now solve for the number of chickens at the beginning of the problem.
2c - 413 = ½(c - 19)

2(2c - 413) = 2[½(c - 19)]

4c - 826 = c - 19

4c - 826 - c = c - 19 - c

3c - 826 + 826 = -19 + 826

3c = 807

c = 269

If the number of the chickens at the beginning of the problem was 269, the number of ducks was twice that, or 538. If the number of ducks at the beginning of the problems was 538, the number of ducks at the end is 413 less than that, or 125.

To check our work, if the number of chickens at the beginning of the problem was 269, the number of chickens at the end is 19 less than that, or 250. Since 250 is twice 125, the number of chickens is now twice the number of ducks.

You Know You're a Geek When You're IM'ing about Factoring Polynomials

Two posts in two days? Pass the smelling salts!

I've been busy writing what amounts to a fifty-page research paper on the teaching of mathematics. It has been eating up much of my time and distracting me from my blog, and I do it gladly. On top of that, Google has decided to delete old videos next month, forcing me to transfer my blog videos to You-Tube and download my private ones. It's a good excuse to attach labels and revamp the blog. The timing stinks!

Wednesday, I tutored an intelligent young woman taking college algebra. She was struggling with factoring polynomials, so we spent over an hour working through problems. My primary goal in tutoring math students is to put them on a search for meaning. If they understand why they are doing what they are doing, they will be more likely to remember it. Even if they forget, meaning helps them reason their way through a problem. If I sniff any hint of wavering, I will ask them why something is true.

My friend had to solve 9x⁵ - 9x³ = 0. She had no problems with step one, dividing both sides of the equation by 9 to get x⁵ - x³ = 0 and knew to pull out x³. When asked what x⁵ divided by x³ was, she made the fatal mistake. She raised her eyebrows and questioned, "X squared?"

Me: "Are you sure about that?"

Her: "Our teacher said you subtract."

Me: "Did she explain why?"

Her: "No. She's old school. She just tells us how to do it. She doesn't have time to explain why."

Sigh. She is smart. She is perfectly capable of understanding why. People who disrespect motivated students enough not to explain why bug me. So, we headed down the path of meaning, peeling back her uncertainty until we reached something solid.

Me: "What does x⁵ mean?"

Her: "You times x by itself 5 times. You know, x times x times x times x times x."

Me: "Good. Any time I'm unsure about a procedure, I start thinking about meaning. If you freeze on a test and forget whether to add, subtract, multiply or divide, you can always fall back on meaning. Write it out the long way."

So, she wrote x⁵ ÷ x³ = (x ∙ x ∙ x ∙ x ∙ x) ÷ (x ∙ x ∙ x). Then I showed her how that is just like saying x ∙ x ∙ (x ÷ x) ∙ (x ÷ x) ∙ (x ÷ x). Then her face lit up, "Oh! Then you can cancel and get 0."

Believe it or not, that is a misunderstanding because we throw around words without meaning and precision and end up confusing students. I responded, "No. Lots of students do that. Let's go back go meaning. What does x divided by x mean?"

Her face went blank. Yes, I know I'm a pain, but this is important! Math makes sense when taught properly. So, I peeled the onion back further. I said, "Sometimes, it is easier to think about numbers. What does 5 divided by 5 mean?"

Another blank stare. The way we teach math focuses on doing, not thinking deeply. I explained, "Dividing means putting objects into equal groups. Suppose you had to share 5 cookies with 5 people. How many cookies would each person get?"

Her: "Oh, 1!"

Me: "What if you shared 10 cookies equally with 10 people?"

Her: "They'd each get 1."

Me: "What if you shared a million cookies equally with a million people?

Her: "They'd get 1!"

Me: "Now, let's get back to x divided by x. What does x mean?"

Her: "I don't know."

My friend answered correctly without realizing it. I explained to her that we use x to represent a number we don't know. It's a placeholder that means a number that we don't know. Having placeholders allows us to set up relationships between known and unknown numbers and manipulate them to figure out the unknowns or refine those relationships. I added, "We have a number of objects and the same number of people. We'll call that number x. If we have x pencils to give to x people, how many pencils would each person get?"

Her: "It's 1."

Me: "What is x divided by x?"

Her: "It has to be 1."

Then, everything fell into place, and she understood:

x⁵ ÷ x³ = (x ∙ x ∙ x ∙ x ∙ x) ÷ (x ∙ x ∙ x)

= x ∙ x ∙ (x ÷ x) ∙ (x ÷ x) ∙ (x ÷ x)

= x² ∙ 1 ∙ 1 ∙ 1

= x²

We had to peel the onion for only a few more glitches. My friend said she had a much better understanding. It disappoints me to know how much rote, meaningless instruction is happening in the math world.

Grace in Geometry

Lisa Cadora's blog post on Grace and Learning got me to thinking about David and his checkered past with math. She described the frustrations of teaching herself to crochet a cool, hip accessory and how much more gracious we are with ourselves than with our students. She concludes,

Charlotte Mason said that the only education is self-education. Did she see that grace is necessary for learning, and that we are most graceful with ourselves? If so, maybe it’s not only that we as teachers must create gracious, grace-ful conditions, environments and relationships in which our students can learn, but that we must bring them to be gracious to themselves.

My husband has two engineering master's degrees, and I have one in statistics. For many years, I thought the math gene had skipped my fifteen-year-old, neurotypical son, David. His temperament is very much like that of my father, who has never met a math problem he liked. Teaching David elementary school math frustrated us both. In hindsight, I think I was part of the problem. I think sometimes, if I had shown more grace, we would have shed fewer tears. Fortunately, he finds algebra and geometry a breeze. Was it maturity and a leap in abstract thinking or a more gracious attitude from me?

I think grace in learning might be related to masterly inactivity (wise letting alone). Elements of masterly inactivity include "authority, good humor, confidence, both self-confidence and confidence in the children," which I lacked because I assumed David would always struggle with math like my father. I stopped looking him as a unique person and saw him as a mirror image of my father because they have so many personality traits in common.

Charlotte Mason believed that we should be gracious enough to let children take personal initiative in their work (page 37-38):

In their work, too, we are too apt to interfere with children. We all know the delight with which any scope for personal initiative is hailed, the pleasure children take in doing anything which they may do their own way; anything, in fact, which allows room for skill of hand, play of fancy, or development of thought. With our present theories of education it seems that we cannot give much scope for personal initiative. There is so much task-work to be done, so many things that must be, not learned, but learned about, that it is only now and then a child gets the chance to produce himself in his work. But let us use such opportunities as come in our way.

On the flip side of this coin, we hurt our children by letting them get so frustrated that they develop the habit of tears. I think we also must keep in mind scaffolding, being alert when to step in and support the child and when to step out a la masterly inactivity. The geometry problem above is a great example. David had to figure out the measurement of each angle in the problem, based upon the diagram and information provided. He had to apply the definitions of bisected angles and right angles, the relationship between vertical and supplementary angles, and the sum of interior angles for triangles (180 degrees) and quadrilaterals (360 degrees). What made this problem difficult is that one wrongly calculated angle would create a domino effect of errors.

Applying masterly inactivity, I left David to his own devices. He worked his way through the calculations and figured out the angles for about five shapes before coming to me because the problem stopped making sense. Then, I switched to scaffolding and congratulated him for recognizing when he was stuck. I studied his work and noticed an error. I erased all of the mistakes and highlighted what was correct, and he went back to work. He went back and forth with me several times, getting frustrated at himself for his errors. Rather than joining him in his vent, I told him about Lisa's blog post about giving yourself grace when making mistakes. I even emailed it to him later in the day. I reassured him that the problem really was challenging and got him back on track.

In the last round, he made another little mistake and I decided to put all of the formulas into a spreadsheet to make sure I was on the right track, too. As I built the spreadsheet, I realized how complicated the problem was. At that point, I was so thankful to have read Lisa's post that morning and let grace win the day.