To learn mathematics, historically students had no choice but to memorize fundamental facts and apply memorized algorithms. Since 1995 in the US, all states have adopted standards to govern K-12 mathematics instruction, and in most, standards have de-emphasized memorization and emphasized reasoning based on concepts. This change assumed the brain could reason in mathematics withoutrelying on memorized knowledge. Scientists who study the brain have recently verified this assumption was mistaken. Due to stringent limitations
in working memory (where the brain solves problems), mathematical problem-solving of any complexity requires applying well-memorized facts and procedures. A decade after the implementation of standards in most states, US young adults ranked last in testing in mathematics among 22 nations. Changes are proposed to state K-12 standards, which recent scientific research suggests would substantially improve student mathematics achievement.

https://www.iejme.com/download/designing-mathematics-standards-in-agreement-with-science-13179.pdf

The researchers behind the Science of Math aren’t arguing that there is no place for inquiry-based learning, conceptual understanding, or growth mindset. They simply argue that concepts and procedures should be taught together, that students should be explicitly taught problem solving skills and then pushed to apply and synthesize those skills, by having students explore complex tasks ALONE will not yield deeper understanding. It’s amazing how people think this means going back to rote memorization and straight computation 100% of the time. This is the kind of all-or-nothing group think that plagues every aspect of political debate in this country and prevents real progress. People think there is one right team and they are on it. People think there is one right way to teach, but the reality of teaching human beings is much more complicated.

Also, let’s talk about equity. As math instruction around the country has deemphasized procedural fluency, many students that are successful in math are getting explicit instruction and procedural practice from their parents or tutors outside of school. By not providing that type of instruction and practice at school, we are actually widening the privilege gap for students with parents who have not worked with them or who were not able to afford outside help. Students whose parents have taught them procedural fluency at home while they have gotten conceptual instruction at school have benefitted from progressive educational movements. However, for the students who have only received the conceptual instruction at school and nothing else, the learning gap has widened.

Also, is Blooms Taxonomy completely dead? According to Bloom, higher levels of learning are dependent on having attained prerequisite knowledge and skills at the lower levels. I feel like lately we have been asking kids to demonstrate knowledge at the upper levels of the hierarchy but they are missing major pieces of the lower levels because they have been deemphasized for years. Just my perspective.

Please allow me to say I think that third graders could not solve this problem because it is an algebra question. When I went to school eons ago algebra was reserved for middle school. I think the question writer was likely unaware they were asking third graders to do algebra without prior instruction in algebra? I would love to know more about this.

The correct algebraic rewriting of the question posed to the third graders is not 96-25=71. A correct interpretation of the question is
96= 25 + x
In this instance to solve for x it is appropriate to subtract 25 from 96 to obtain an answer.

I teach physics to high school students and I think that looking ahead to the skills needed for physics down the road it would be of great help to understand why 96-25=75 is not the same thing as 96= 25 + x and that the latter is what was being asked.

The first is how we are defining math improvement. If an assessment measures rote memorization and/or application of mysterious procedures, and measures breadth over depth, then of course it will reinforce the idea that we should keep teaching that way. Then, the researchers can say, “see, it’s better to teach procedures and algorithms.”

That’s like measuring a healthy diet by only looking at how much weight loss it promotes. Sure, weight loss can be a good thing (like algorithms are a good thing for dealing with difficult calculations). But only focusing on weight loss might yield an incredibly unhealthy diet, just as focusing on algorithms might yield a very unhealthy relationship with true mathematical reasoning.

Most of us teachers have seen, over and over, that the same small group of students do well at math (and seem to like it), while the majority either just “get by” or, worse, reject math and mathematical reasoning and say things like, “I guess I’m not a math person”.

It may be harder to teach conceptual math. And, it may be much harder to assess whether students are able to think mathematically and apply that thinking in the real world. But I think we can figure all of that out when we accept that a major change is needed.

The second issue is about math curriculum in general. Why is it that we focus so much on abstract algebraic thinking, trigonometry, and a sort of “all roads lead to calculus” mindset rather than a focus on statistics, correlation / causation thinking, and financial literacy? Most people will never factor polynomials, calculate cosines, or use derivatives in their work or personal lives. But literally everyone will make financial decisions and literally everyone will deal with statistics in some way (for example, political decisions, news, analyzing what school to send their kids to…). Why aren’t we making math curriculum decisions that create more capacity for true mathematical reasoning that can be applied to the real world in a concrete way? Ironically, that strategy of real-world math might help so-called “non-math students” embrace math, and therefore be up for the challenge of factoring polynomials, converting graphs into linear equations, and/or finding sine, cosine, and tangents.

Instead, for most students, math is a mysterious set of rules and procedures that yield an answer. Oh, and to check that answer, go ahead and apply a different set of mysterious procedures. It’s insane! How can anyone think this is good? That’s not the type of math activities mathematicians do. That’s not what scientists do when they think about how to collect and analyze data. That’s not what stock analysts do. Accountants…ok maybe. But why shouldn’t our students get the chance to have the same experience as the scientists and mathematicians who created those awesome algorithms and theorems? Then, when they know why the algorithms or theorems work, they can embrace those procedures for the efficiency they bring to harder calculations.

This shift in strategy wouldn’t preclude students from taking Calculus. People often want to use “but…” statements. Like, “well, I like this idea, BUT when will they have time to do pre-calculus?” First off, there should always be options for students who love so-called “higher math”. Even so, I would argue that even the high-achieving students would benefit tremendously from more statistics and financial literacy, which are chalk full of critical thinking and mathematical application. In other words, even the current “winners” of the math game would benefit from a math reasoning approach.

I realize that another common reaction to these ideas is a statement like: “but to get into a good college, they need to at least get to pre-calculus in high school,” or, “well, the system just isn’t set up for this”. Right, that is the structure we have, but if we all agree that structure could be improved, then changes can start to happen. Sea changes are slow, but rely on all of us being honest about what we see in our classrooms, in our own children struggling with math, and with the lack of mathematical reasoning we see out in the world.

Or, we can just sit on our hands and keep the status quo, and keep wondering why the same silent majority dislikes math, why they often don’t use math reasoning effectively in their personal lives, and why so many say, “I’m just not a math person”. And those of us who are “good at math” can keep promoting more of the same instead of being open to a more holistic, authentic, mathematical reasoning approach.