mathematics Archives • https://educationalrenaissance.com/tag/mathematics/ Promoting a Rebirth of Ancient Wisdom for the Modern Era Sun, 30 Apr 2023 02:17:50 +0000 en-US hourly 1 https://wordpress.org/?v=6.8.3 https://i0.wp.com/educationalrenaissance.com/wp-content/uploads/2025/04/cropped-Copy-of-Consulting-Logo-1.png?fit=32%2C32&ssl=1 mathematics Archives • https://educationalrenaissance.com/tag/mathematics/ 32 32 149608581 The Habit of Reading: Five Book Recommendations for 2023 https://educationalrenaissance.com/2023/01/21/the-habit-of-reading-five-book-recommendations-for-2023/ https://educationalrenaissance.com/2023/01/21/the-habit-of-reading-five-book-recommendations-for-2023/#respond Sat, 21 Jan 2023 12:00:00 +0000 https://educationalrenaissance.com/?p=3493 It’s January of a new year! And so you are probably inundated with a number of calls to implement new habits, to try new practices, and to start new programs. Hopefully this list of recommended reading for 2023 cuts through the noise and provides you with at least one great read for the upcoming year. […]

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It’s January of a new year! And so you are probably inundated with a number of calls to implement new habits, to try new practices, and to start new programs. Hopefully this list of recommended reading for 2023 cuts through the noise and provides you with at least one great read for the upcoming year.

C.S. Lewis, The Abolition of Man

I begin with a book that rivals in many ways the essay by Dorothy Sayers that got our educational renewal movement started. In fact, C. S. Lewis delivered these lectures (the Riddell Memorial Lectures were a series given over three nights at King’s College, Newcastle University on 24–26 February 1943) a good four years before Sayers (her paper was read at the Vacation Course in Education at Oxford University in the Summer of 1947). If you have read “The Lost Tools of Learning,” then you are well prepared to tackle these essays.

In three essays, Lewis mounts a defense of objective value in the face of moral subjectivism. He predicted the dystopian future we now live in where tolerance is the reigning virtue, despite the fact that we are not a very tolerant people, at least one wouldn’t think so when one reads comments on social media. This book provides a foundational rationale for the “classical” part of our movement. (This book pairs nicely with Mere Christianity, connecting the “Christian” part of our movement.) And yet it nicely goes beyond what we might consider a fixation on Western civilization as the sole or sufficient basis for a liberal arts education. We see this most prominently in his use of the Tao as representative of objective values based on natural law. What he is getting at transcends an East/West divide and demonstrates that values are meta-cultural.

Sample Quote: “This things which I have called for convenience the Tao, and which others may call Natural Law or Traditional Morality or the First Principles of Practical Reason or the First Platitudes, is not one among a series of possible systems of value. It is the sole source of all value judgements. If it is rejected, all value is rejected. If any value is retained, it is retained. The effort to refute it and raise a new system of value in its place is self-contradictory. There has never been, and never will be, a radically new judgement of value in the history of the world. What purport to be new systems or (as they now call them) ‘ideologies’, all consist of fragments from the Tao itself, arbitrarily wrenched from their context in the whole and then swollen to madness in their isolation, yet still owing to the Tao and to it alone such validity as they possess. . . . The rebellion of new ideologies against the Tao is a rebellion of the branches against the tree: if the rebels could succeed they would find that they had destroyed themselves. The human mind has no more power of inventing a new value than of imagining a new primary colour, or, indeed of creating a new sun and a new sky for it to move in.”

C. S. Lewis, The Abolition of Man (Harper, 2000): 43-44.

I could see this book being valuable if you are a teacher or administrator. It is also well worth adopting in an upper-level humanities course.

If you would like an opportunity to delve deeply into this book, there is an upcoming event you might consider joining if you are located in the American mid-west. The Alcuin Fellowship will be meeting on March 30-April 1 at Clapham School in Wheaton. We’ll be reading The Abolition of Man and having rich discussion around the book in small groups. There are limited spaces available. You can register for this fellowship at https://www.alcuinfellowship.com/midwestern-alcuin-retreat-2023/.

Jonathan T. Pennington, Jesus the Great Philosopher

Okay, so I reviewed this book in two posts back in the autumn of 2021. Jonathan is a good friend, and this is a good book. I keep returning to it because it offers such a compelling synthesis of Christianity with the liberal arts tradition. The wisdom of this book abounds, and we benefit repeatedly from the insights of a leading New Testament scholar. Yet, Pennington also puts the cookies on the bottom shelf, so to speak.

This book goes well with the previous selection, although it offers a more modern mix of metaphors and imagery. There’s a brilliance in being able to bring such individuals as Aristotle and Steve Martin together as Pennington does. I think you’ll find this is a volume that can speak to teacher and student alike.

Sample Quote: “Hence, as we have seen throughout this book, there is insight to be gained from what the philosophers said about all sorts of topics. We needn’t cut ourselves completely off from their wisdom. Rather, we can gather lumber from whatever trees are available as we build the Christ-shaped temple of our lives, with Holy Scripture as the building inspector. As Justin himself said, “Whatever things were rightly said among all men, are the property of us Christians. . . . For all the writers [ancient philosophers and poets] were able to see realities darkly through the sowing of the implanted word that was in them. For the seed and imitation that is imparted according to capacity is one thing, and quite another is the things itself, of which there is the participation and imitation according to the grace which is from Him.”

That last part gets a bit complex, but the point is straightforward – any wisdom in the world is from God, who created all, but we Christians have the grace that enables complete understanding. This includes the grandest human philosophical question: What does it mean to live a whole, meaningful, and flourishing life? What is the wisdom we need for the Good Life?”

Jonathan T. Pennington, Jesus the Great Philosopher (Brazos, 2020): 203.

Barbara Oakley, A Mind for Numbers

My next selection moves away from the humanities and provides something for those STEM teachers among us. Having taught Geometry for several years, I have appreciated how Barbara Oakley spells out effective learning strategies for students. I myself was never a great math student, and diving into teaching math well over a decade ago required going back to the basics. Along the way I found that math itself is not particularly difficult, but it can be quite different than the kinds of learning that goes on in the humanities side of the curriculum.

Oakley bases her work in solid neurological studies. One of the key insights in her book is to “chunk” mathematical and scientific concepts. A chunk is a conceptual piece of information that is “bound together through meaning.” (54) That “meaning” bit is significant because there’s a sense of the personal importance. The chunk attracts information or ideas to it, providing for mental leaps as separate units of information bind together through neural networks.

She provides three steps to forming a chunk. First, you focus your attention on the information to be chunked. (57) She advises learning in a low-distraction environment, free from screens. One of the core concepts here is that old neural networks enable you to form new neural pathways. In other words, we build from the known to the unknown. In essence, we want to create these chunks off of ideas, concepts or information that we already know well.

Second, you need to understand the basic idea (58). She differentiates the initial moment of understanding – the “aha!” moment – from the kind of understanding where you can close the book and test yourself on the problem. This is very much the way narration works. Being able to bring forward the formula, the steps, or the process in mathematics demonstrates that the idea is understood.

Third, you need to connect the basic idea to a context (58-59). In other words, a student needs to know when, say, apply the Pythagorean theorem, and when not to. She likens the chunk to a tool, “If you don’t know when to use that tool, it’s not going to do you a lot of good.” (59)

Chunking is not only valuable in mathematics, but across the curriculum. You can chunk historical concepts or literary terms. Chunking can be a pathway toward integration as we allow that chunk to attract more and more concepts to it. I think this is similar to Charlotte Mason’s expression about ideas, “Ideas behave like living creatures––they feed, grow, and multiply.” (Charlotte Mason, Parents and Children, 77)

Sample Quote: “A synthesis – an abstraction, chunk, or gist idea – is a neural pattern. Good chunks form neural patterns that resonate, not only within the subject we’re working in, but with other subjects and areas of our lives. The abstraction helps you transfer ideas from one area to another. That’s why great art, poetry, music, and literature can be so compelling. When we grasp the chunk, it takes on a new life in our own minds – we form ideas that enhance and enlighten the neural patters we already possess, allowing us to more readily see and develop other related patterns.”

Barbara Oakley, A Mind for Numbers (Tarcher Perigee, 2014): 197.

What I like about this book is that her strategies are not simply about how to test better to get good scores on tests or entrance into college, etc. Instead, she sees how this can be a pathway to deep meaning in life through acquired skill, and how an individual can achieve creativity in multiple domains of knowledge through accumulated competence. The quote comes from a section entitled “Deep Chunking,” which segues nicely to our next book.

Cal Newport, Deep Work

Associate professor of computer science at Georgetown, Cal Newport not only delivered a best-selling book, but coined a phrase that has become part of the cultural parlance: “deep work.” In many respects, this is a counterpoint to Nicholas Carr’s The Shallows inasmuch as Newport accepts the premise that the internet has made us shallow and then goes on to propose a solution by going deep through focused attention. The book is designed in an interesting way. Newport begins by spelling out three ideas that get at the “why” of deep work. Then the second part of the book spells out the “how.” Here I want to focus on the first part.

Newport’s first two ideas interact with the new economy centered around knowledge work: deep work is valuable largely because it is rare. This points to a “market mismatch” where talented individuals who are able to produce knowledge that is deep. His third idea is that deep work is meaningful. This is an idea that riffs on the metaphorical meaning of the word “deep.” When our work connects to something of the human experience, there’s a depth of character that has intrinsic value. I like how Newport develops the concept of craftsmanship as a sacred practice.

Sample Quote: “Once understood, we can connect this sacredness inherent in traditional craftsmanship to the world of knowledge work. To do so, there are two key observations we must first make. The first might be obvious but requires emphasis: There’s nothing intrinsic about the manual trades when it comes to generating this particular source of meaning. Any pursuit – be it physical or cognitive – that supports high levels of skill can also generate a sense of sacredness.”

Cal Newport, Deep Work (Grand Central, 2016): 88-89.

As our skill increases, our sense of the meaning we are generating also increases. One gets plugged into the creative impulse that is part of our own imago Dei createdness. Now this is a point that is likely remote from Newport’s thinking, but his use of the word “sacred” points in this direction. Newport goes on to explain his second key observation that to access this deep meaning, we must embrace deep work as the portal to cultivating our skill.

One of the reasons why I recommend this book is that it has provided a framework for understanding how our educational renewal movement – perhaps counterintuitively – gives our students a strategic advantage as they enter the new economy. By encountering the deep ideas of the great works our students get connected to a level of depth not present in the school system. Many of our schools feature intense instruction on writing and rhetoric, which is essential to the knowledge work Newport describes as so rare and valuable. Graduates from classical schools are well trained to do deep work. So, by reading this you can cultivate the habit of deep work in yourself and your students.

Gerald Graff and Cathy Birkenstein, They Say / I Say

My final selection is a textbook ostensibly for college writing. This year I adopted this title for our junior rhetoric class. It is full of practical advice for writers learning how to build effective arguments in academic writing. We are using the fifth edition, which came out in 2021, but any of the editions that have come out since the original 2006 edition features most of the same contours.

The central idea of the book is that effective argumentation begins with a good understanding of what others have said before venturing into an expression of one’s own beliefs. They posit that “working with the ‘they say / I say” model can also help with invention, finding something to say. In our experience, students best discover what they want to say not by thinking about a subject in an isolation booth but by reading texts, listening closely to what other writers say, and looking for an opening through which they can enter the conversation.” (xviii). As classical educators, we are very aware that the great books tradition is all about the great conversation. How better to take advantage of the plethora of books we read than by utilizing that conversation to initiate new pathways for our students to explore based on the “they say / I say” model.

Another feature of this book is how it utilizes templates. The authors recognize the liability of training students to use templates. “At first, many of our students complain that using templates will take away their originality and creativity and make them all sound the same.” (13) But through practice and instruction, students begin to see how there is a basic structure to how good argumentation works. Even after initial exposure to these templates, we can analyze academic writing to identify not only the basic “they say / I say” structure, but also finer points of perspective, argumentation, and analysis. For students raised on the 10-sentence paragraph and the five-paragraph essay, this approach to templates builds on earlier types of templates.

Students are able to practice utilizing two major questions as they work through this book. There is the establishment of relief (using an idea from sculpture), between what you are proposing and what others might say. Students begin to become sensitive to the question, “Oh yeah, who says otherwise?” The other question that students learn to become aware of is the “so what?” or “what difference does this make?” set of questions. For students in junior rhetoric, this is excellent training for the work they will accomplish the following year during senior thesis. The essential skills students learn in this book are critical analysis of sources, summary of conventional viewpoints, handling controversial topics, and expressing the application and consequences of one’s point.

One chapter I really appreciate is the chapter on revision. For many students, revision amounts to identifying typographical errors and eliminating the teacher’s red marks. Well, the approach taken by the authors provides a handy guide to how to make substantial revisions to an essay.

Sample Quote: “One of the most common frustrations teachers have – we’ve had it, too – is that students do not revise in any substantial way. As one of our colleagues put it, “I ask my classes to do a substantial revision of an essay they’ve turned in, emphasis on the word ‘substantial,’ but invariably little is changed in what I get back. Students hand in the original essay with a word changed here and there, a few spelling errors corrected, and a comma or two added. . . . I feel like all my advice is for nothing.” We suspect, however, that in most cases when students do merely superficial revisions, it’s not because they are indifferent or lazy, as some teachers may assume, but because they aren’t sure what a good revision looks like. Like even many seasoned writers, these students would like to revise more thoroughly, but when they reread what they’ve written, they have trouble seeing where it can be improved – and how. What they lack is not just a reliable picture in their head of what their draft could be but also reliable strategies for getting there.”

Gerald Graff & Cathy Birkenstein, They Say / I Say (Norton, 2021): 149.

After this introduction, which describes what many a teacher has felt, the authors provide guidance on how to make substantial revisions to an essay. The chapter on revision concludes with an excellent revision checklist. Students regularly run into the same frustrations we have with revision. They have a sense that they could express their thoughts in a better, more sophisticated way, but they are unpracticed in how to excavate their own writing with a view to finding the veins of gold, let alone finding the weaknesses to correct.

Conclusion

Hopefully this list of books to read in 2023 will inspire you to dig into some different areas where you can become a more inspired and skilled educator this year. There are tons of other books I could have recommended, and you likely have some of your own that are top of your list.

Even more essential than reading the selection of book listed here is building the habit of daily reading. Even a little bit on a daily basis begins to accumulate to a significant amount of input into your life. With lesson planning, grading, meetings and family life, it can be difficult to carve out time to read. Steven Covey talks about how important it is to “sharpen the saw.” For us educators, reading is one of the best ways for us to cultivate the joy of learning we want to inspire in our students. So whether it’s these books or others that spark interest in you, take a moment even now to read.

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True Mastery: The Benefits of Mixed Practice for Learning https://educationalrenaissance.com/2020/01/04/true-mastery-the-benefits-of-mixed-practice-for-learning/ https://educationalrenaissance.com/2020/01/04/true-mastery-the-benefits-of-mixed-practice-for-learning/#comments Sat, 04 Jan 2020 12:40:29 +0000 https://educationalrenaissance.com/?p=791 “Practice, practice, practice.” This mantra for learning is proclaimed across companies and schools, athletics and the arts. The widely held belief is that the key to mastering a particular skill or gaining new knowledge is relatively straightforward: Practice.  Now, to be sure, practice is important, especially if it rises to the threshold of “deliberate practice,” […]

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“Practice, practice, practice.” This mantra for learning is proclaimed across companies and schools, athletics and the arts. The widely held belief is that the key to mastering a particular skill or gaining new knowledge is relatively straightforward: Practice. 

Now, to be sure, practice is important, especially if it rises to the threshold of “deliberate practice,” an intensive approach which Patrick lucidly explained in a past article. He himself warns, however, that the repeated rehearsal of skills can be futile if the three other components of deliberate practice are not in play. Patrick writes,

“ We need to be careful with this first component. It is all too easy to set up high frequency and think we are accomplishing something, when in fact all we are doing is a long series of empty work.”

Here it is acknowledged that merely bumping up the frequency of practice is not enough to hone a skill or understand a concept, particularly complex ones that are multifaceted and layered.

Applying this to the classroom, what can be done to ensure that the sort of practice our students engage in is not wasted? From differentiating between direct and indirect objects to solving algebraic equations to writing thoughtful, well-developed essays, how can we train our students in such a way that the skills they develop and knowledge they gain remain in their memories long-term? 

The Myth of Massed Practice

In Make It Stick: The Science of Successful Learning, authors Peter Brown, Henry Roediger, and Mark McDaniel propose that one of the keys for long-term, strong and flexible mastery of a skill or concept is to mix up the practice. But first they dispose of the myth of massed practice. Massed practice is the focused, repetitive practice of one thing at a time until it is mastered (47). To be fair, this approach to learning is fairly intuitive and therefore enjoys quite a bit of trust from onlookers and practitioners alike. For example, the common advice for increasing one’s free-throw percentage in basketball is to shoot over and over again from the fifteen-foot mark (the distance from the free-throw line to the hoop). For those of us with basketball experience, we can even testify to the success of this strategy, specifically how quickly we experienced gains by utilizing this method. 

But real learning, insists the authors of Make It Stick, includes more than how quickly the skill is mastered or knowledge is acquired. What matters even more is whether that skill or knowledge is accessible to our memories when it is needed later:

“The rapid gains produced by massed practice are often evident, but the rapid forgetting is often not” (47).

The litmus test for successfully learning a skill, then, is whether this practical knowledge remains in our memory long-term. Scientists call the increased performance during the acquisition phase of a skill “momentary strength” and distinguish it from “underlying habit strength” (63). 

I have experienced this distinction between momentary strength and underlying habit strength too often in my ongoing development as a handyman. Thanks to the internet, when something in my house needs fixing or replacing, I need only search for the relevant video online to find out how to solve the problem. After watching the video (ten times over) and then completing the job myself, I gained momentary strength in the skill. I walked away from the project feeling confidently handy. However, three months later, when the same problem returned, I discovered that in the previous experience I had not actually gained underlying habit strength. As I scanned my memory for knowledge and proficiency of the skill, the search results were clear: no records found. (Note: I suppose I am exaggerating. My memory could recall some bits and pieces of what I had learned previously. But it wasn’t nearly sufficient for what I need to fix the problem again. Too much had been lost.)

Why? I had used massed practice, developing momentary strength, but not underlying habit strength that would serve me well long-term. I watched the video, spent a focused, inordinate amount of time honing the skill, and then neglected to practice it for three-months. Over this duration of time, the practical knowledge I gained was lost. It faded away into the distance, along with other short-term memories, like what I wore to work that day or what I had for breakfast.

So failing to gain long-term retention is one problem with massed practice, but even worse, this approach to learning, according to researchers, generally leads to inflexible, surface-level comprehension that is not amenable for complex cognitive acts like differentiation and application. In other words, it doesn’t lead to true mastery. 

Three Ways to Mix it Up

In order to reach this level of true mastery, mixing up the practice is the way to go. And, according to the latest research, there are three main ways to do so: 

1. Spaced Practice: Instead of intensively focusing on one skill for a single, extended session, space out the practice into multiple sessions. Work on it for a while and then return to it the next day or week. The space allotted between each session allows for the knowledge of the skill to soak into one’s long-term memory and connect to prior knowledge. Revisiting the skill each session certainly takes more effort, but that’s the point:

“The increased effort required to retrieve the learning after a little forgetting has the effect of retriggering consolidation, further strengthening memory” (49).

2. Interleaved Practice: While we experience the quickest gains by focusing on one specific skill or subset of knowledge over a period of time (momentary strength), interleaving, or mixing up the skill or concept you are focusing on in a practice session, leads to stronger understanding and retention (underlying habit strength). It feels sluggish and frustrating at times, for both teachers and students mind you, because it involves moving from skill to skill before full mastery is attained, but it leads to both a depth and durability of knowledge that massed practice does not (50).

3. Varied Practice: Varying practice entails constantly changing up the situation or conditions in which the skill or concept is being applied. It therefore strengthens the ability to transfer learning from one situation and apply it to another, requiring the student to constantly be assessing context and bridging concepts. Because this jump from concept to concept triggers different parts of the brain, it is more cognitively challenging, and therefore encodes the learning “…in a more flexible representation that can be applied more broadly” (52).

Test Case: Mixed Practice in Math Class

Let’s apply this concept of mixed practice to math class. The teacher is explaining to her students how to calculate the volume of geometric figures. The “massed practice” approach would have her teach the formula for calculating the volume of, say, a cube and then release her students to find the volume for ten different sized cubes. By the end of the class period, the majority of her students would be comfortable performing the algorithm, thereby demonstrating ostensible mastery of the skill (at least in the short-term).

Contrast this scenario with the “mixed practice” approach. The teacher might start by teaching the formula for the volume of a cube and then giving her students a couple practice problems to solve. But soon after, she would teach a different formula, say, the formula for finding the volume of a cylinder and, after that, a sphere. Upon giving her students a couple practice problems for each type of figure, she would instruct her students to complete a set of problems, in which various volume problems are interleaved. This problem set would force students to practice discernment: identify the particular figure, recall the relevant algorithm, and then run the numbers. Periodically, over the following days and weeks, the teacher would include various volume problems in the class warm-up and in the daily homework as well as other concepts previously covered.

As you can see from this example from math class, when mixed practice is implemented, underlying habit strength is forged. The order and types of exercises don’t permit a student to fall into mindless, rote practice. Instead, each problem requires higher level thinking skills beyond memorization such as categorization and application. In order to gain true mastery, the research is clear: mix up the practice. 

Mixed Practice and the Liberal Arts

Let me leave readers with one final thought: I’ve been using the modern phrase “true mastery” as shorthand for the sort of breadth and depth of learning we are aiming for at our schools and in our homes. But I could just as easily describe this outcome using language from the liberal arts tradition. When students engage in the challenge and rigor of mixed practice, they are being trained to learn and think for themselves, to engage in a form of self-education, which is a central idea in the classical tradition (as Jason explained so eloquently in his recent article for Circe). Their tools of learning, the liberal arts, are being sharpened, so to speak.

In addition, by providing opportunities for our students to engage in mixed practice, we avoid teaching them bad habits of cramming, that is, surface-level mastery for brief demonstration on an upcoming test. Instead we give them an opportunity to perform dynamic and valuable work, stretching their minds in flexibility, durability, and discernment, all of which is befitting of their God-given intellects and capabilities.

This is the sort of learning I get excited about and hopefully through this blog more parents and teachers can join us at Educational Renaissance in this life-giving work. 

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