Formative Assessments Dept.

(A preview/snippet from my book “Out on Good Behavior: Teaching math while looking over your shoulder”)

Some say summative assessments can be used formatively, by using the results to guide approaches in subsequent courses. The overlapping nature of how these definitions have evolved give me much cover in my quest to appear aligned with the edu-party line. During my first year at Cypress, I allowed my classes to use notes for quizzes, but not tests. I felt that this would reinforce the idea of the value of notes. The problem was that some students’ organizational skills were lacking—resulting in this typical conversation:

Student: How do you do this problem?

Me: Look in your notes.

Student: I can’t find it.

Me: (Drawing a diagram on a mini-white board.) How would you find the time each of the cars are driving?

Student: I don’t know.

Me: (Writing “Distance = Rate x Time” underneath the diagram)

Student: Oh!

I knew that there was a potential that such approach could quickly blossom into grade inflation and an artificial sense of achievement. So I justified my giving them help by telling myself “Well, I guess this is a formative assessment and I’m using the results to guide future instruction.” But I knew there were limits.

Math Zombies and Conceptual Understanding in Math


Conceptual understanding in math has served as a dividing line between those who teach in a conventional or traditional manner (like myself), and those who advocate for progressive techniques. The progressives/reformers argue that understanding of a procedure or algorithm must precede the procedure/algorithm itself; failure to do this results in what some call “math zombies”.

I will state that I, like many teachers, do in fact teach the underlying concepts for algorithms, procedures and problem solving strategies. What I don’t do is obsess over whether students have true understanding nor do I hold up a student’s development when they are ready to move forward.

For many concepts in elementary math, understanding builds from procedures. The student practices the procedure until it is realized conceptually through familiarity and tactile experience that forges pathways and connections in the brain. (Efrat, 2018). Daniel Ansari (2011), maintains that procedures and understanding provide mutual support. Sometimes understanding comes first, sometimes later. And I’m fine with that.

There is No One Fixed Meaning to Understanding
What does understanding mean? Does it mean to know the definition of something? In freshman calculus, students learn an intuitive definition of limits and continuity which then allows them to learn the powerful applications of taking derivatives and finding integrals. It isn’t until they take more advanced courses that they learn the formal definition and theorems of limits and continuity. Does this mean that they don’t understand calculus?

Does understanding mean transferability of concepts? Or, as a teacher I had in Ed school put it: “What happens when students are placed in a totally unfamiliar situation that requires a more complex solution?” Dan Willingham, a cognitive scientist who teaches at University of Virginia calls being able to transfer knowledge to new situations “flexible knowledge”. Willingham (2002) explains that it is unlikely that students will make such knowledge transfers readily until they have developed true expertise. He argues, “[I]f students fall short of [understanding], it certainly doesn’t mean that they have acquired mere rote knowledge and are little better than parrots.” Rather, they are making the small steps necessary to develop better mathematical thinking. Simply put, no one leaps directly from novice to expert.

How Do You Test for Understanding?
One proxy that teachers use for understanding and transfer of knowledge, is how well students can solve problems and their variations. A student in my seventh grade math class recently provided an example of this. As an intro to a lesson on complex fractions, I announced that at the end of the lesson they would be able to do the following problem:

The boy raised his hand and said “Oh, I can do that now; I know how to solve that.” He then narrated what needed to be done. He had certainly never seen this exact same problem before. And while he did not know why the invert and multiply rule worked, he put together basic skills that he learned and saw how they fit together and solved a more complex problem—an example of knowledge transfer.

Is Understanding Always Necessary to Solve Problems?
When does understanding help in solving problems or doing procedures? In my experience, it does when the concept is part and parcel to the procedure. An example: knowing what procedure to use to simplify 𝑥2 ∙ 𝑥5 versus(𝑥2)5. Students often have trouble remembering when exponents are added and when they are multiplied. The concept of multiplying powers is helpful; in the first case, the student remembers it is(𝑥 ∙ 𝑥 ) ∙ (𝑥 ∙ 𝑥 ∙ 𝑥 ∙ 𝑥 ∙ 𝑥), and it is easily seen that the exponents are added. In the second case, raising a power to a power, the same principle applies: (𝑥2)5 = 𝑥2 ∙𝑥2 ∙𝑥2 ∙𝑥2 ∙𝑥2 which lends itself to understanding that the exponent “2” is multiplied by 5.

When the concept is not as closely attached to the procedure, (e.g., some trigonometric identities) the conceptual underpinning may not be as accessible. In such cases, the understanding may not necessarily help to solve problems.


Ending the Fetish over Understanding

While some basic levels of understanding are thought of as “rote memorization”, lower level procedural skills inform higher level understanding skills in tandem. Reform math ignores this relationship and assumes that if a student cannot explain in writing a process used to solve a problem, that the student lacks understanding and is a math zombie.
As a former college football player and high school football coach told me recently:

“Worrying about math zombies is like worrying that your football players are too good at passing the ball — on the basis that their positional play is no better than the rest of the team, and therefore they obviously don’t understand what they are doing when they pass
beautifully.”

Given that today’s employers are complaining over the lack of basic math skills their recent college graduate employees possess, the math reform movement has created a poster child in which “understanding” foundational math is often not even “doing” basic math.

Shameless Self Promotion, Dept.

For actual examples of hands-on, real-world experiences of understanding vs procedure as it happens in the classroom, then read my book “Out on Good Behavior” before it’s ruined by the Hollywood movie version. Plus it has the intrigue of school politics wrapped in the enigmatic axiom of “You never really know for sure what’s going on.”

References
Ansari, D. (2011). Disorders of the mathematical brain: Developmental dyscalculia and mathematics anxiety. Presented at The Art and Science of Math Education, University of Winnipeg, November 19th 2011

Furst, E. (2018) Understanding ‘Understanding’ in blog Bridging (Neuro)Science and Education https://sites.google.com/view/efratfurst/understanding-understanding?authuser=0

Willingham, D. (2002) Inflexible knowledge: The first step to expertise, in American Educator 26, no. 4 7 (2002): 31–33, 48–49. https://www.aft.org/periodical/american-educator/winter-2002/askcognitive-scientist

FAQ’s about “Out on Good Behavior”

As I’ve mentioned, my new book is out and available. To elucidate and amuse future readers, (and taking a cue from the Common Core website) I’ve compiled some Frequently Asked Questions about my book.

Is “Out on Good Behavior” about the Zen of teaching math?

Nope. Just the usual rebelling against the edu-fads and how I make it look like I’m on board with the current educational lunacy.

You talk about two students who you helped on an intervention basis and you state that qualifying as a special needs student doesn’t guarantee the student will get the kind of help to deal with a disability.  Can you elaborate?

There are students who are classified as “special needs” under the IDEA law. In most cases, particularly in math, they are given accommodations such as extra time on tests, and an aide to help explain, or even to take notes for the students. But for students who may suffer from the various forms of “dyscalculia”—i.e., inability to memorize key facts and procedures, inability to think abstractly—they more often than not need the help of a specialist. Getting extra time on tests is not going to solve memory and other problems. In the end, they may continue to be given remedial classes, and to learn and re-learn the same things over and over again. In the end, however, they generally make little progress.

Is this what you tried to do with the JUMP Math curriculum?

In a sense. I had a class of seventh graders who had deficits in math knowledge. The advantage of JUMP is that it breaks things down into small increments of knowledge that students can absorb and build upon. It helped some of my students build confidence in their ability. It was not a silver bullet; some of them still could have benefitted from a specialist. But it was a step in the right direction.

You mention your use of the 1962 algebra book by Dolciani. Do you ever get complaints from parents about your use of that book?

Any and all reaction from parents about the Dolciani book has been positive.  One parent told me “This is how I learned algebra, and I’m able to help my daughter.”  Others like the simplicity of the format and the problems.  I also hear from the students who like it because “It doesn’t have those real-world problems.”

Teaching is a second career for you; something you took up after retiring from the work force.  Do you ever regret not getting into teaching when you were younger?

No, because I probably would have been swayed by the ed school dogma that pervades education. Being older I was able to resist. I know others who changed careers as I did and who say the same thing. My experience and age allow me to trust myself to do things differently.

Do you think that there are some students who won’t achieve the “understanding” that is being pushed so heavily?

Without a doubt. There are some things that people will understand later—like how the point-slope formula really works to find the equation of a line, or why we invert and multiply when dividing fractions. Repeating a procedure, particularly when more mathematical tools are learned helps in that regard. But yes, there are some who may never understand.

Will those who never understand do worse in math than those who do understand?

Not necessarily. It depends what they are doing in life.  To use an example from calculus, the definition of limits and continuity are quite formal. Those who major in math and who wish to become mathematicians need to understand how they work. Those who go on to become engineers will not be less qualified to do what they do. They are able to reap the fruits of what limits and continuity do mathematically; i.e., they can find derivatives and do integration, and solve complex engineering problems. Not fully understanding the theory behind these things will not interfere with their work.

Please feel free to send more questions. Or buy the book. Or both!

As You Haven’t Been Told, Dept.

I teach math at a small Catholic school in California.  I teach 7th grade math and 8th grade algebra.  For those who have read my latest book, you know that I use a 1962 version of Dolciani’s “Modern Algebra” as my textbook.  The students like the simplicity of its presentation, and so do the parents. I have had parents tell me they like the book, and one in particular said that it is how she learned algebra, and it allows her to help her daughter. She thanked me, and said “I can’t stand that Common Core stuff.”

The Common Core stuff that irks parents are the alternative “strategies” that replace the standard algorithms. One of these is for multidigit addition/subtraction; another is for multiplication and division. As I’ve documented before, these strategies (such as ‘making tens’) are nothing new. Traditional textbooks of the past have taught them, but they were introduced after students mastered the standard algorithms.  The standard algorithms served as the “main dish” in the dinner party known as math. The alternatives were “side dishes” and the two were distinguishable. Now they are not; it is one big mess, with students sometimes thinking that they have to use a particular strategy for particular problems.

Therefore it is of interest to hear William McCallum’s view of this aspect of Common Core. He was one of the two lead writers of the Common Core math standards.  When I wrote an article that was published in the online Atlantic about Common Core, I pointed out that the standard algorithm for multi-digit addition and subtraction did not appear until 4th grade. Until then, teachers and students were saddled with “strategies” which included pictures and inefficient methods in the name of “understanding”. The view of reformers is that teaching standard algorithms first eclipses the conceptual underpinning of why the algorithms work as they do—this in spite of the pictorial explanations that appeared in early textbooks from the 60’s, 50’s and earlier that provided such explanation.

McCallum commented on my Atlantic article and disagreed with me that the standard algorithms were delayed. I provided him evidence until he finally stated that the Common Core standards do not prohibit the teaching of the standard algorithms prior to the grade in which they appear.  Specifically his comment was:

The standards (1) do not say that conceptual understanding must come first, and (2) also say explicitly on page 5 that ‘These Standards do not dictate curriculum or teaching methods. For example, just because topic A appears before topic B in the standards for a given grade, it does not necessarily mean that topic A must be taught before topic B. A teacher might prefer to teach topic B before topic A, or might choose to highlight connections by teaching topic A and topic B at the same time.”

This was news to me, and apparently news that was buried in the material accompanying the standards, despite McCallum’s belief that it was made clear.  In particular, a guidance document for publishers, which came out in tandem with the Common Core standards, advises publishers not to test students on standard algorithms prior to the grade in which they appear in the standards.  I guess it’s OK to teach the standard algorithms earlier, but just not to ensure that students know them.

There are few “Common Core aligned” textbooks that address the standard algorithms prior to the grades in which they appear in the Common Core standards, so apparently McCallum’s word has not really made the rounds.  There is one exception and that’s the Common Core editions of Singapore Math. They do teach the standard algorithms earlier. They also test the students on them, which has cost them a penalty by the company EdReports which rates textbooks in terms of the degree to which they are aligned with Common Core.  Singapore Math’s Common Core edition is considered by EdReports to not be aligned.

Since people mistakenly believe that alignment with Common Core implies effectiveness, EdReports’ rating of Singapore’s books may have cost the company some sales.

The Standards continue to be interpreted in accordance with math reform ideology. And although McCallum in his remarks to me in the comment section of the Atlantic article stated that “the phrases ‘critical thinking’ and ‘collaborative learning’ do not occur anywhere in the standards and that the standards “neither dictate nor forbid any particular style of pedagogy”, the die has been cast for the lower grades (K-6).

In the world of Common Core, alignment equals effectiveness, and book publishers happily comply with the guidelines they have been given. In my opinion, as well as others, it would have been helpful if McCallum’s statements to me could have been made more public than in the comment section of an Atlantic article. It also would be helpful if publishers are not punished by outfits such as EdReports for ensuring that students know standard algorithms prior to the grade in which they appear in Common Core.

Side Dish vs Main Dish, Dept.

My previous post about solving problems in multiple ways has been interpreted in multiple ways. In previous eras, students were taught the standard algorithm first. The standard algorithms for multi-digit addition and subtraction were explained via diagrams and other means to show what was really happening when we “carry” and “borrow” (now called “regrouping”). I.e., it was not taught without understanding.

After mastery of the standard algorithm, students were shown alternative methods such as “making tens” and other short-cuts, which spotlighted the conceptual underpinning behind the standard algorithm. Often, students discovered these methods themselves. Anchoring mastery with the standard algorithm first created a distinction and students could see what was the main dish versus the side dish.

Current procedure under textbook interpretations of Common Core, is to delay teaching the standard algorithm, and to teach the alternative methods first under the belief that the standard algorithms eclipse understanding. In so doing, the distinction between main and side dishes are obscured and students are often confused, sometimes being forced to solve problems by inefficient methods such as drawing pictures when they are clearly ready to move on.

Two quotations from Steve Wilson, math professor at Johns Hopkins come to mind in this regard from an article that appeared in Education Next.

There will always be the standard algorithm deniers, the first line of defense for those who are anti-standard algorithms being just deny they exist. Some seem to believe it is easier to teach “high-level critical thinking” than it is to teach the standard algorithms with understanding. The standard algorithms for adding, subtracting, multiplying, and dividing whole numbers are the only rich, powerful, beautiful theorems you can teach elementary school kids, and to deny kids these theorems is to leave kids unprepared. Avoiding hard mathematics with young students does not prepare them for hard mathematics when they are older.

There will always be people who think that you must be able to solve problems in multiple ways. This is probably similar to thinking that it is important to teach creativity in mathematics in elementary school, as if such a thing were possible. Forget creativity; the truly rare student is the one who can solve straightforward problems in a straightforward way.

Then there are those who say that “if only I had been taught why these algorithms work”, now seeing the math through an adult lens. They fail to see that their ability to understand may have come from being taught in a traditional method that did, in fact, teach the conceptual understanding, but without the same degree of obsession about it that now pervades math education.

Solving Problems in Multiple Ways, Dept.

There is a notion floating about in education land that teaching students multiple ways to solve particular mathematical problems builds flexible thinking, and reasoning skills. I have been looking for research studies that show this, but the closest I’ve found is a study by Rittle-Johnson et al (2016) conducted in algebra classes, that had students compare and discuss alternative methods. It does not address the effect of learning multiple methods for solving a problem. And it also does not definitively conclude that the comparison of methods increases flexible thinking and reasoning skills.

Flexible thinking comes up often in edu-land because it is associated with a nagging question—one that was articulated to me by my advisor when I was attending ed school:  “What happens when students are placed in a totally unfamiliar situation that requires a more complex solution? Do they know how to generate a procedure? How do we teach students to apply mathematical thinking in creative ways to solve complex, novel problems? What happens when we get off the ‘script’?” 

This question has been addressed in part by cognitive scientist Dan Willingham who argues that if students fall short of solving novel problems “it certainly doesn’t mean that they have acquired mere rote knowledge and are little better than parrots. There is a broad middle-ground of understanding between rote knowledge and expertise.” Simply put, no one leaps directly from novice to expert.

 While there is no direct path to learning the thinking skills necessary to apply one’s knowledge and skills to unfamiliar territory, Willingham argues that one way to build a path from inflexible to flexible thinking is through worked examples. Students extend their knowledge along scaffolding built from examples—examples that fit over the underlying structure. Although it does not necessarily happen automatically, thinking becomes more flexible as more knowledge and experience are acquired. 

The current interpretation of the seventh grade Common Core Math Standards as it applies to ratios and proportion provides a case in point. One of the authors of the standards, Phil Daro, was apparently guided by an unmoving and unshakeable conviction that traditionally taught math was nothing more than “getting the answer”.  He has spoken about proportional reasoning and how it has been taught with no regard to process or conceptual understanding. I suspect that he is the main reason why proportional reasoning is now taught with multiple methods. 

To put this in perspective, those who were taught “the old way” remember problems that asked to solve problems like “If John can type 100 words in 2 minutes, how many words can he type in 6 minutes? Students then solved, using the equation 100/2 =x/6.

The problem could be done in two ways. The first was cross multiplication, obtaining 2x = 600, and x = 300 words.  The other way was multiplying 100/2 by 3/3 to get the equivalent fraction 300/6, which immediately revealed that 300 words could be typed in 6 minutes.

Cross multiplication, in the eyes of Daro and others with similar reform math inclinations, is viewed as a “trick” that obscures the conceptual understanding, even though the process is based on sound mathematical principles. That is, if a/b = c/d, it is easy to see that multiplying both sides by the common denominator of bd, results in ad=bc, thus explaining why cross-multiplication works.   And, I may add, that those principles are taught to students, (usually using numbers instead of letters to cut down on abstractness). Students tune this out, in general; they are more interested in doing the problem. Despite the resulting student confidence in their problem solving, cross multiplication is still looked upon as a “trick” and “rote procedure.”

To thus counteract what is perceived as rote memorization, students are now taught that they can solve the problem by finding the unit rate first, and then multiplying. In the above problem, then, the unit rate is 50 words in one minute. Multiply by six to find the number of words typed in six minutes.

Having taught this method to seventh graders, I see some students confused: “Which way do we do it” and “When do we use unit rate and when do we use the other way with cross multiplication?”  But the purveyors of multiple methods have thought of this, so they have extended it even further. Let “w” equal words typed and “m” equal minutes. Then students are taught to express “w” divided by “m”, or w/m as the unit rate. In the above problem, we would have w/m=50.  Solve for “w” to obtain w = 50m, and voila! A formula! Now we can find out the words typed for any number of minutes by plugging into a formula. And they don’t have to use “w” and “m”, they can use “y” and “x” which gets to the next extension of ratio and proportion: direct variation.

Now students learn that equations in this y = kx form are called “direct variation”. And they can be graphed! And the graph goes through zero, and is a straight line!Then they are taught what slope is, and taught that “k” is the slope, which is the same as “unit rate” which is also called “constant of proportionality”.

I have taught these lessons for several years now and can tell you that seventh graders are immensely confused. Those inclined toward progressive math approaches would say that they’re confused because I am teaching it wrong.

And I agree. I am teaching it wrong.

Because to teach it right, you should just teach the basic proportion equation with cross multiplication and leave it at that like it used to be done. Once a student has something that works every time and they have confidence, then they can branch out and explore other possibilities. In particular, when they take algebra later, they can build upon mastered foundations, adding richness through other representations as the contexts present themselves, such as tables, graphing and slope. In this manner, they are motivated to learn other ways of looking at a familiar problem.

I would agree that it makes good pedagogical sense in having students solve things in more than one way. Demanding it as a necessary element of instruction can cause cognitive overload, however. As math professor Rob Craigen says, “Overemphasis may lead not to an ability to think outside the box, but for the box to be lost.”

Reference: Rittle-Johnson et al., (2016) “Comparison and Explanation of Multiple Strategies: One Example of a Small Step Forward for Improving Mathematics Education” in Policy Insights from Education Research, Volume 3 Issue 2, October

A “must read” by a former math teacher

Ted Nutting wrote this piece, which is worth reading, remembering, and passing around the internet:

 In the one year that I taught a course for which there was a state end-of-course test (Algebra 1 in the 2011-2012 school year), my students scored better than those from any other teacher in the district.  I have the data to prove all this. Why did this happen?  I broke the rules and taught real math.  In calculus, I used a textbook more aligned with real teaching than the book I was supposed to be using.  In algebra, not having an alternative textbook, I made up my own worksheets to accompany the lessons I gave.  I actually taught.  I presented the material, asking questions frequently to keep students’ attention, and I gave difficult quizzes and tests.  I demanded good performance — and the results were excellent. 

Articles I never finished reading, Dept.

In an Education Week compilation devoted to “Start the Year With a ‘Primary Focus’ on Relationship-Building” there are several articles, none of which I could finish reading. Here are excerpts from two of them: The first is by Melanie Gonzales, an elementary math curriculum, advanced academics, and early-childhood coordinator in Texas.

“Based on the work of Carol Dweck and Jo Boaler, teachers will encourage students to build a growth mindset. Additionally, time will be spent reminding students that mathematicians notice things, are curious, are organized self-starters, and effective communicators and problem solvers. Finally, they will use their math skills to count out a specific number of snack items and celebrate being mathematicians already!”

The second is by Emily Burrell, a mathematics teacher and co-lead mentor teacher at South Lakes High School in Fairfax County, Va.:

“I teach high school mathematics students who have been marginalized by the public education system. Traditional teaching methods have failed them. It may not be surprising that many of them have failed a math class. My students are uninspired to do math that doesn’t matter to them. I reach these students by providing a curriculum that does matter: a project-based curriculum that provides choice and helps students build their voice.”

See if you can do better than I did.

More of the Same, Dept.

A recent article in “Smart Brief” argues that if you change parents’ attitudes about math, you will change the childrens’. This makes sense, but the devil is in the details as they say. The study the author describes (and which she conducted) to substantiate this, views the changing of parents’ attitudes as educating them in the alternative strategies that students are forced to learn in lieu of the standard math algorithms. The standard math algorithms are now delayed until 4th, 5th and 6th grades per the prevailing interpretation of Common Core–and the textbooks that put this interpretation into practice.

The starting thesis for the article is as follows:

“Many parents’ beliefs about effective mathematics instruction are inconsistent with current research.”

Depends what “current research” you’re looking at I guess. I wouldn’t know reading this article, because the author doesn’t cite any. She refers to parents’ attitudes toward the Common Core math standards as a “misunderstanding”. Interesting choice of words. I’d say that it’s probably a case that the people who think the “understanding uber alles” approach of the Common Core math standards is effective, is a misunderstanding. A misunderstanding about what understanding in math is about.

“Parents try to explain computation the way they learned it a generation ago. Children partially learned a different strategy or algorithm earlier that day but can’t put all of the pieces together. They can’t make sense of the procedural-based traditional algorithm parents are showing them. Parents can’t make sense of the concept-based algorithm or invented strategy the child is showing them. The session often ends in tears.”

What the article doesn’t choose to say is that the standard algorithms that the parent teaches their frustrated children generally works well. She says the opposite–they can’t make sense of it. Characterizing the standard algorithms as something the students can’t make sense of is inaccurate. And the standard algorithms for multiplication, division, addition and subtraction can be explained (and were in the older textbooks) in terms of their conceptual underpinnings.

As far as what the author refers to as “concept-based algorithms” or “invented strategies” (which the students likely didn’t invent but had them thrust upon them by a CC-aligned textbook), these are nothing new. They were taught also in earlier eras, but after the standard algorithms were taught and mastered. There were strategies like “making tens” or adding from left to right. For example 56 + 79 can be done by adding 50 + 70 (or 120) and 6 + 9 )or 15). The partial sums are added to get 120+15 or 135.

Ironically, some of these techniques were sometimes discovered by the students themselves. Now, however, it is a mish-mash of these techniques, taught to ensure that students “understand” what is happening with place value. The belief is that teaching the standard algorithms first obscures the conceptual understanding.

Adding to that students’ confusion, they are also required to make drawings of what is going on, in the belief that “visualizing” the math is understanding. What results is confusion of a plethora of techniques, like a dinner of side dishes. The standard algorithms do not stand out as main dishes, but just another side dish and they often are left wondering which side dish would be most appropriate for the problem at hand.

I had an algebra student who had to multiply two two-digit numbers. He used a convoluted partial products technique that took up much space on his paper and which he had trouble doing. I tried to show him the standard algorithm, but the habits were set and it was just more confusion.

The thrust of the “study” the article examines is that by educating parents (and pre-service math teachers) in the alternative methods and strategies, it boosted parents’ confidence as well as their children’s. I would like to see the study, Actually, I wouldn’t. I’ve seen similar ones. They lack control groups in general, and contain an inherent confirmation bias.

The author, Carol Buckley, is identified as an associate professor of mathematics at Messiah College in Pennsylvania. I looked her up. She has a B.S. in Elementary Education and an M.Ed. in Curriculum and Instruction from Shippensburg University; and an Ed.D in Educational Leadership from Immaculata University. But no degrees in math.