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.

New Boss Old Boss, Dept. (Covid 19 edition)

In light of the rapidly approaching school year, there have been a host of articles about how teaching must change.  And so I was not terribly surprised to see that National Council of Mathematics Teachers (NCTM) and the National Council of Mathematics Supervisors (NCSM),have jumped on this bandwagon and announced that math teaching must change in their latest report.

An article summarizing NCTM’s report states: “According to the NCTM and NCSM, during the pandemic, the urgency to change the way mathematics is taught has become apparent. According to both agencies, math instruction needs to be more equitable, so it is essential to plan what math classes will look like before returning to school in the coming months.”

Reading through the article, as well as the NCTM/NCSM document itself, other than the fact that online teaching by its nature is different than in-class teaching, it is not apparent how mathematics must be taught differently. In fact, the NCTM/NCSM document’s advice on how math should now be taught is generally the same as it has been for the past three decades. Namely “differentiated instruction”, elimination of ability grouping, full inclusion, and equity for all.

Their pleas for these changes make it seem as if nothing in math education has changed in the past thirty years. If anything, there has been an increase in the practices so recommended. Elimination of ability grouping has been accomplished by so-called differentiated instruction by providing different assignments and expectations for the varying levels of student abilities within the same class.  The teaching of procedures and algorithms has given way to “understanding and process”.  A disdain for memorization has de-emphasized the learning of multiplication tables. The teaching of standard algorithms is delayed while students learn inefficient and confusing “strategies” that purportedly show the conceptual underpinning behind the standard algorithms.

The document advises that specific teaching practices be implemented in online learning.  The document then provides eight practices that the authors of this document believe provide equitable and effective math teaching, and which “provoke students to think.”

Here they are with my commentaries attached:

• Set math goals that focus on learning.

How else are math goals established? The implication, given NCTM’s past history, is that providing instruction for procedures, with worked examples and scaffolding is “inauthentic” and therefore is not focused on learning.

• Implement tasks that promote reasoning and problem-solving.

Most textbooks that were written in previous eras did just that, and did it well.

• Use and link mathematical representations.

By this they mean students should be able to visualize what’s happening by means of pictures. Also, they want students to make “connections” with prior mathematical topics. Robert Craigen, a math professor at University of Manitoba who has been involved in improving K-12 math education says this: “It’s amusing when they speak about “connections” as if this were something different from “isolated facts”.  Actually it is the facts that provide connections.  Everything else is only the educational analog of a conspiracy theory.”

• Facilitate meaningful problem-solving course.

They want problems to be “relevant”, in the belief that otherwise students have no desire to solve them. Actually, students will want to solve problems for which they have been given effective instruction that allows them to be successful at it.

• Ask questions with a purpose.

This could refer to “intentionality” or “math talk”, or both. Let’s look at “intentionality” first.

Inentionality is the edu-buzzword du jour which has replaced the previous one: “student agency.” From what I can tell from its usage, “intentionality” generally means an overriding goal that strongly colors—and drags along—all other considerations of a lesson. So if the goal is differentiating the lesson to take into account the “variability of all learners”, then any other goals for a particular lesson—say multiplying negative numbers—must be constructed to accommodate weak students and challenge stronger ones.

Math talk: This refers to getting students to talk “like mathematicians” by asking questions such as “Can you convince the rest of us that your answer makes sense?” and “What part of what he said do you understand?” I recently saw an article claiming that “research shows” that students who talk about their math thinking are motivated to learn. In addition, this “math talk” is viewed as a form of formative assessment giving teachers a peek into student thinking and where they need help.  “Math talk” is an effective tool only if the instruction they received allows them to make use of it. Otherwise, it is like children dressing up in their parents’ clothes to play “grownups”.

• Develop procedural fluidity that comes from conceptual understanding.

Although they pay lip service to procedural fluency, it is fairly clear that they believe that mastery of the conceptual understanding behind a procedure must always precede the learning of said procedure.

• Support the productive struggle in learning mathematics.

Worked examples with scaffolding are believed to be “inauthentic” and take away from what would otherwise be a productive struggle. Missing from this type of reasoning is that a person who is trying not to drown is not learning how to swim.

• Obtain and use evidence of students’ mathematical thinking.

In other words, students must be able to explain their answers. While this can be done through questioning, it does not take into account that novices (particularly in lower grades) are not as articulate as adults think they should be. Adults have had many years of experience with the topics that novices are trying to learn.  “Show your work” now means more than showing the mathematical steps one does to solve the problem. It means justifying every step. Failure to do so, even if a student has correctly solved a problem is viewed as the student failing to “think mathematically” or understand.

I’ll leave it to you to read the NCTM/NCSM document in its entirety. In all fairness, some of their advice is useful.  But in my opinion most of it is not.

“Problem solve” Dept

A recent article announced that the National Science Foundation (NSF) funded a grant for West Virginia University College of Education and Human Services The grant is to help educate math teachers on a new way of teaching math to teachers. For those of you new to all this, NSF spent billions of dollars in grant money in the early 90’s to fund (in my opinion and the opinion of many others) ineffective and damaging math programs including Investigations in Number, Data and Space; Everyday Math; Connected Math Program; Core Plus; and Interactive Math Program.

Of particular interest to me was this sentence: “The hope was for math teachers to find ways to teach students how to problem-solve.”

It used to be that students solved problems.  But now in today’s era of math reform, they “problem-solve”. Popular use of this rather irritating verb form harkens back to NCTM’s 1989 standards which downplayed the importance of procedural skills, and replaced those with students achieving “deeper understanding” and being able to problem-solve.

The core belief behind the current math-reformers’ use of the term “problem-solving” is that it is a core competency that can be taught independent of the domain in which a problem appears. Little to no importance is given to mastery of procedural skills, instruction on how to solve particular types of problems, nor sufficient practice solving such problems.

The typical problems of the past (distance/rate, mixture, number, coin) are being replaced with what reformers believe are problems that students are interested in wanting to solve. These are typically one-off problems that don’t generalize and for which little to no prior problem solving procedure has been taught. 

One math reform approach has been to present students with a steady diet of “challenging problems” that neither connect with the students’ lessons and instruction nor develop any identifiable or transferable skills. The following problem from Hjalmarson and Diefes-Dux (2008) is one example: How many boxes would be needed to pack and ship one million books collected in a school-based book drive? In this problem the size of the books is unknown and varied, and the size of the boxes is not stated.

While some teachers consider the open-ended nature of the problem to be deep, rich, and unique, students will generally lack the skills required to
solve such a problem, skills such as knowledge of proper experimental approaches, systematic and random errors, organizational skills, and validation and verification.  The belief is that just as students develop problem solving habits for routine problems, a similar “habit of mind” or problem-solving schema occurs for solving non-routine problems.

Based on my experiences as both student and teacher, as well as the experiences of veteran math teachers, I submit that a substantial education in mathematics should steer a middle course between the proliferation of routine problems and reliance upon unique, complex projects. Students should learn to apply basic principles in a much wider variety of situations than typically presented in texts. Such problems, however, should not be as
complex or as time consuming as the example above. A math problem is not necessarily useful just because it requires outside-of-the-box insight and/or inspiration and will generally not result in a problem-solving “habit of mind” or schema.

Problem solving techniques taught independent of the domain in which they occur include such things as “work backwards”, and “find a simpler but similar problem”. But without experience, practice and mastery of domain-specific problems, asking a student to find a simpler but similar problem is as useful as telling a novice bike rider to “be careful” when taking a ride on their own.

Sweller et al. (2010) state that problem solving cannot be taught independently of basic tools and basic thinking. Over time, students build up a repertoire of problem-solving techniques. Ultimately, the difference between someone who is good and someone who is bad at solving nonroutine problems is not that the good problem solver has
learned to solve novel, previously unseen problems. It is more the case that, as students increase their expertise, more nonroutine problems appear
to them as routine.

Looks like the idea of problem solving as a core competency will be taught to a bunch of lucky teachers in West Virginia thanks again to the misguided largess of the National Science Foundation.

References

Margret A. Hjalmarson and Heidi Diefes-Dux (2008), Teacher as designer: A framework for teacher analysis of mathematical model-eliciting
activities, Interdisciplinary Journal of Problem based Learning, Vol. 2, Iss. 1, Article 5. Available at http://dx.doi.org/10.7771/1541–
5015.1051.

John Sweller, R. Clark, and P. Kirschner (2010), Teaching general problem-solving skills is not a substitute for, or a viable addition to, teaching mathematics, Notices of the American Mathematical Society, Vol. 57, No. 10, November.

Cheerleaders for Common Core, Dept.

Because of school closings due to Covid 19, there has been a flurry of articles about distance learning, and the difficulties that parents face when having to explain “Common Core” math. The articles take the opportunity to show that parents are just not “with it” and that the new way is actually better because it confers “deeper understanding” rather than rote memorization.

This article is typical as is the following quote from it:

“Amberlee Honsaker remembers learning only one way to add or subtract in elementary school. It was the standard algorithm: stack numbers vertically, add the digits in columns, and carry the ones where necessary. For her daughter, Raegan, math instruction extends far beyond that. In first grade, Raegan is using number bonds, making place-value charts, drawing out 10s and ones  — illustrating multiple methods for solving simple addition problems.”

Actually, in my elementary school as well as for many others, there were alternate methods taught. But they were taught after mastery of the standard algorithm. The alternate methods in addition to being taught were also often discovered by students themselves as an outgrowth of the mastery of the standard algorithm.  A problem like 76 + 85 could be solved by adding 70+80 to get 150, and then 6 +5 to get 11. Adding 150 and 11, the final sum of 161 is obtained. 

Number bonds were called “fact families” and place value charts were abundant as a glance through textbooks from the 60’s, 50’s, 40’s, and further back easily show. (See this article for examples)

But now, alternate methods are taught first in the belief that it imparts a “deeper understanding” of what is going on with standard algorithms and procedures which are taught later. Teaching the standard algorithm first is thought to obscure the understanding and is viewed as a “rote” procedure. As a result, what is mischaracterized as “rote memorization” has been replaced with “deeper understanding” as math reformers term it. I think a more accurate term is “rote understanding”.

 The so-called “deeper understanding” is measured by having students show more than one way to add or multiply numbers, and to explain in writing why it works.

From the article:

“Over the past 40 years, education research has emphasized that teaching math should start with building students’ understanding of math concepts, instead of starting with formal algorithms, according to Michele Carney, an associate professor of mathematics education at Boise State University.”

The article does not do us the favor of providing us references to the research but I’ve seen some of it. Most of it is based on “action” research done in classrooms with questionable controls, and authored by the same people who have been taking in each others’ laundry for years. (e.g. Fenema, Carpenter, Hiebert, etc)

Common Core codified much if not most of the reform math ideology that has been at work for more than three decades.  Reform ideology got its first big boost with NCTM’s math standards in 1989 which was predicated on the notion that traditional math teaching sacrificed conceptual understanding on the altar of procedural fluency. It put an emphasis on “understanding” and viewed procedures as nothing more than “rote memorization”.

The other catch-phrase of the math reformers is “problem-solving”; so much so, that it has become a verb. It used to be that students solved problems.  Now they “problem-solve”. Again, this harkens back to NCTM’s 1989 standards which downplayed the importance of procedural skills, and replaced those with students being able to “explain” their answers. “Math talk” has emerged as an indicator for whether students “understand”.  If a student cannot explain how they solved a problem, they are held to lack understanding. Also, if a student cannot solve a problem in more than one way, that too is held to show a lack of understanding. 

The typical problems of the past (distance/rate, mixture, number, coin) are being replaced with what reformers believe are problems that students are interested in wanting to solve. These are typically one-off problems that don’t generalize and for which little to no prior problem solving procedure has been taught. 

The “problem solve” mentality has made its way into ed schools where I heard the philosophy espoused. That is, there is a difference between problem solving and exercises. “Exercises” are what students do when applying algorithms or problem solving procedures they know. Problem solving, which is preferred, occurs when students are not able to apply a mechanical, memorized response, but rather have to figure out what to do in a new situation. Moreover, ed school catechism states that students’ difficulty in solving problems in new contexts is evidence that the use of “mere exercises” or “procedures” is ineffective and they are overused in classrooms.

It is more likely that students’ difficulty in solving new problems is because they do not have the requisite knowledge and/or mastery of skills—not because they were given explicit instruction and homework exercises.

Those who make such a differentiation and champion “true” problem solving espouse a belief in having students construct their own knowledge by forcing them to make connections with skills and concepts that they may not have mastered. But, with skills and concepts still at a novice level, students are not likely to be able to apply them to new and unknown situations. Nevertheless, the belief prevails that having students work on such problems fosters a discovery process which the purveyors of this theory view as “authentic work” and the key to “real learning.”  One ed school professor I knew summed up this philosophy with the following questions: “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’?” 

In fact, as Rittle-Johnson, et al. (2015) have shown, procedural fluency does not exclude conceptual knowledge—it can ultimately lead to conceptual understanding. Also, “Aha” experiences and discoveries can and do occur when students are given explicit instructions, worked examples, and scaffolded problems.

While some educators argue that procedures and standard algorithms are “rote”, they fail to see that exercising procedures to solve problems requires reasoning with such procedures — which in itself is a form of understanding. This form of understanding is particularly significant for students with LD, and definitely more useful than requiring explanations that students do not understand for procedures they cannot perform.

Reference:

Rittle-Johnson, Bethany; Michael Schneider, Jon Star “Not a one-way street: Bidirectional relations between procedural and conceptual knowledge of mathematics.” Educ. Psychol Review; DOI 10.1007/s10648-015-9302-x