I stumbled on this article about Australian math education, and I see a lot of similarity with US education. The big difference is they seem to have systems that operate province-wide, so one uninformed bureaucrat can have a big impact. The US system is highly fragmented, so you never know what you are going to get. Some excerpts: One of its fundamental failings is the treatment of algebra, by which measure the new curriculum falls short by a country mile. Algebra is the beating heart of mathematics. It is the naming of the quantity being hunted, setting the stage for its capture. It is how we signify pattern and how we express the relationship between quantities. You want to understand something else, probability or statistics or geometry? Algebra is essential. Algebra is how Descartes captured geometry, and how Newton and Leibniz captured calculus. A fundamental insight is that algebra is simply arithmetic, just with numbers we don’t know. The current curriculum states this clearly, in a Year 7 instruction to ‘Extend and apply the laws and properties of arithmetic to algebraic terms and expressions’. By contrast, ACARA’s new curriculum instructs Year 7s to ‘generate tables of values from visually growing patterns or the rule of a function; describe and plot these relationships on the Cartesian plane’. This is pointless, and it is not algebra, except in the most trivial sense. It is substituting data entry and graphical busywork for the critical practice of algebraic skills. The curriculum is choked with such nonsense, strangling the few stray instructions that might otherwise engender proper study. The term ‘algebra’ and the critical discipline of algebra have been distorted to meaninglessness. ********************** There is, to recoin a phrase, no royal road to algebra. A mastery of algebra requires practice and memorisation and struggle. And preparation. Prior to the arithmetic of numbers we don’t know comes the arithmetic of numbers we do. In order to understand a/b = c/d, we must first understand 4/6 = 6/9. This is the work of primary school. ********************** The treatment of mental skills is little better. Automatic recall of the multiplication tables is critical to all the arithmetic that follows: a student cannot apply 4 x 9 if they are struggling simultaneously to remember it or calculate it. The multiplication tables were mangled in the draft curriculum and then fixed, but only up to a point, where they are referred to as ‘multiplication facts’.9 This is weird and telling. Much worse, the tables (still) only reach 10 x 10, rather than the traditional 12 x 12. This matters. We have 60 minutes in an hour and 360 degrees in a circle for a reason. Having natural multiples and factors and fractions readily at hand is critical for the learning of arithmetic. The ignorant decision to exclude 12 turns out to not matter, however, since few students even get to 10. The large majority of primary school students do not learn their multiplication tables. This is because, clear curriculum direction notwithstanding, the very large majority of primary school teachers do not consider the tables important, much less mandatory. Teachers have been fed other ideas. ********************** ACARA’s new curriculum is a large but entirely predictable step down. Modern education is steeped in grandiose perversion, with innovative but misconceived practices working not to improve but to undermine fundamental processes of understanding. There is Higher Order Learning, and 21st Century Skills, and Flipped Classrooms, and Child-Centred Learning, and Discovery Learning, and Inquiry Learning, and Play Based Learning, and on and on and on. Underlying almost all of this is the philosophy of constructivism. Built upon the tautology that we only understand what we understand, constructivism claims that students must construct this understanding through their own experience. Constructivist approaches are to be contrasted with the boring old practice of directly teaching students, and in particular the teaching of clear facts and skills. Always lurking is the boogieman of Rote Learning. This boogieman in particular has frightened teachers away from orderly ‘tables’ and into embracing isolated ‘facts’. There is a proper and important role for mathematical exploration, but that role in primary education is limited. It took thousands of years for civilisation to come up with the crystal concepts and truths and techniques of ‘elementary’ mathematics. Constructivism is the slowest and most painful and least successful method of mastering these fundamentals. Not everyone, however, sees this as a drawback. A teacher focused on students’ Higher Order Thinking may have little concern for the lowly basics. Mistakenly. Rather than basic facts and skills being opposed to deeper thinking, the basics are the foundation for deeper thinking. Before twenty-first century skills, whatever these might be, there must come a mastery of seventeenth-century skills. ********************** Hand in hand with constructivism goes problem-solving, one of the great con jobs of mathematics education. Mathematicians love the idea of problem-solving, since it is a one-word definition of what they do. Mathematicians love to see children working on authentic, well-structured problems, with clear mathematical content and purpose. But school problems are different. School ‘problems’ are typically poorly defined, open-ended explorations with no measure of or concern for success, and with students ill-prepared for a venture of any significance. These problems are all the worse for almost invariably being about something other than mathematics, about a largely fictitious Real World. Thus, rather than a carefully crafted problem about prime numbers and factors, students ‘explore’ or ‘model’ the painting of walls and the graphing of mortgage rates. Students are presented with pseudo-problems that require little thought and inspire less. ACARA and the education authorities regard the Real World as a great selling point. Hence the new curriculum has a stream on ‘Space’ rather than geometry. Hence NAPLAN has a test on ‘Numeracy’ rather than arithmetic. Hence the mandating of statistics way beyond its very limited pedagogical worth. Hence the unceasing focus on STEM, which reduces mathematics to an instrumentalist, utilitarian skill set. This all fits in well enough with the neoliberal notion of education as training, but it is otherwise reductive, and it erodes the basis for mathematical thinking. Real Worldness in school is almost invariably contrived, and thus as boring as dirt, because we simply need very little mathematics for our everyday lives. Such Real Worlding also feeds back to devalue and to poison the teaching of mathematics. The critical point of mathematics, the source of its incredible power, is that it removes the distracting noise of the world. Mathematics abstracts from messy reality to create something much simpler, something that can be analysed and honed and generalised. And yes, mathematics then gives back, providing indispensable tools for the understanding of real-world phenomena. But the mastery of these tools is beyond the scope of school mathematics for any but the most banal of real-world situations. What results is simply the glorification of noise—the presentation of noise as the central topic of mathematics education. It is absurd, and disastrous. ********************** Mathematics education wastes untold time and energy and goodwill on electronic media: students watch videos instead of reading; they ‘move’ shapes on screens instead of shifting physical blocks; they push calculator buttons instead of computing on paper; they ‘prove’ statements by pressing Solve or Graph on their handheld computers. ********************** Plenty are fooled by constant references to ‘visual learners’ and ‘digital natives’, but there is no fooling reality. The perverting effect of these media is that students are not required to think or to reflect. They need never pay proper attention, to a teacher or even to their own thoughts. The electronic media stimulate and entertain, occupying the space where contemplation might have occurred. In his 1986 book Amusing Ourselves to Death, Neil Postman wrote on the effect of television on education: ‘The name we may properly give to an education without prerequisites, perplexity and exposition is entertainment’. That is where we are, except that now television, and much worse, are in and are intrinsic to the classroom. ********************** THE PAST IS A FOREIGN COUNTRY And the name of that country is Singapore. For instance. Asian countries dominate TIMSS, the international test of school mathematics. Asia even dominates PISA, the anti-algebraic non-test created specifically so that Western countries would feel better about themselves. Even if one wishes to concentrate upon ACARA’s snake-oil games, it turns out that attention to the basics is the way to go. Australia once did much better, although that was long ago and is largely forgotten. The powerful forces of entertainmenting have been at work for many decades. Moreover, there are two intertwined forces, one political and one philosophical, which have directly perverted mathematics education. ********************** First, the political. Historically, for good and bad, Australian mathematics education was carefully controlled by education bureaucrats under the guidance of mathematicians.15 In the 1970s, teachers started to be given more autonomy.16 Also around that time, a fourth group, of education academics, was beginning to emerge as a force.17 Since then, and with varying overlaps and alliances, these four groups have tussled over the nature and control of mathematics education.18 All-out war broke out in the early 1990s, with the bureaucrats attempting to wrest control from the other three groups.19 The bureaucrats failed, but the downward slide was well underway. The power of education academics has continued to grow, and they are now much more closely wedded to the bureaucrats. ********************** With mathematics education academics now in possession of their own world, they are generally much less connected than they once were to the world of mathematics; they are less adept at and less interested in it. This lack of mathematical expertise encourages and necessitates an emphasis on other, non-mathematical concerns, laying the fertile ground for constructivist obfuscation. Much more time is spent in apologising for and avoiding the difficulty of mathematics than is ever spent addressing that difficulty, or in demonstrating the beauty and the power that can result from proper effort.

Another upvote for MCT, especially his grammar and vocabulary threads. Rules of grammar are clearly explained and stuck with my kids all through high school. The Caesar's English vocab is delightful. We dropped the poetics thread, but now I wish we hadn't.

I also found this helpful document on writing proofs. This particular professor sounds like he doesn't care for my approach of inserting symbols into the text which is fair. A good reminder to ask your professor when in doubt.

For the sake of her graders, she may want to write in non-cursive handwriting, and learn some symbolic conventions and abbreviations, if approved by your instructor. If your student is thinking about math for graduate school, she may want to learn Latex and get comfortable using Overleaf. For example, \rightarrow yields this symbol: which can be used in place of "implies" or "it follows that..." \in yields: which can mean "Is a member of this set" but I often use this liberally outside of sets for my own convenience Page 1 of this document has more symbols that may her proofs easier to read. Also, does she center her important equations on their own line? Does she line up her equals signs and include a lot of white space? That will help improve readability.

@GoodnightMoogle yes yes YES! Teachers participating in Project Follow Through (the single largest educational study in the world) also hated the program because it was highly scripted and the teachers were micromanaged to a fault. Probably what was happening is the teachers just weren't that smart...aka, they lacked 'domain-specific knowledge.' But they came around when they saw for themselves how effective it was, and how well their students performed. I remember people advocating for "Direct Instruction" on these boards, and I thought it was a straw man argument. I mean, who out there is actually advocating for NOT teaching the students? Turns out the straw man is real. While I think Rate My Professor is a useful tool, I will also argue that students are often not the best judge of a good teacher, kind of akin to Dunning Kruger. I will admit I prefer a teacher who is entertaining, like stand up comedians. I may not be a good judge of my teachers until 5-10 years later.

Prompted by this thread I reviewed all my kindle notes and highlights from the Barton book. Here they are in a disorganized mess. Enjoy: “Memory is the residue of thought, so students remember what they think about.” Dressing up as Pythagoras does not help your students remember the pythagorean theorem. “Observing engagement alone does not imply learning. The problem is, of course, what exactly were they engaged in? Unless we have further evidence (a test of retention, for example), we must be extremely careful in concluding that learning is taking place.” Kids talking and debating each other can only confuse them. “Domain-specific knowledge makes thinking and learning easier, and the automation of knowledge further reduces the strain on working memory.” Spend more time doing math. Making math appear relevant to real life may backfire. “Very rarely did these contexts lend themselves perfectly to the maths I wanted to teach. Often I would need to present a modified, simplified version that bore little resemblance to the original context. Moreover, not all my students found these contexts useful or even that interesting. Students are constantly on their guard against being conned into being interested. By attempting to appeal to students’ interests we risk excluding those students who do not share those interests. I have concluded real-life maths is often more trouble than it is worth.” “make sure all the students are aware of this, so they see that everyone struggles and it is perfectly normal to make mistakes.” “Without Apology: ‘embrace – rather than apologise for – rigorous content, academic challenge, and the hard work necessary to scholarship’.” “Simply immersing students in a series of unstructured maths tasks is not likely to enable them to learn the fundamentals of the topic. most children will not be sufficiently motivated nor cognitively able to learn all of [the] secondary knowledge needed for functioning in modern societies without well organized, explicit and direct teacher instruction. Precious time would be spent discussing and debating, correcting and re-explaining – time that could have been spent practising and applying our new-found knowledge.” “questioning the practice of letting students devise their own methods to solve problems: ‘What is left unsaid is that when a child makes up an algorithm, the act raises two immediate concerns: 1) whether the algorithm is correct 2) whether it is applicable under all circumstances under a less guided approach these mistakes and misconceptions are harder to identify and hence deal with appropriately, with students working on many different things.” “Students remember what they think about or attend to. By giving students such freedom we reduce the chances of them attending to the things that really matter. Siegfried Englemann’s model of Direct Instruction outperformed all other models in basic skills. However, what is perhaps most interesting is that Direct Instruction also outperformed all other models on measures of problem-solving and self-esteem. Resting this responsibility on the shoulders of novice learners risks jeopardising their learning, and that is simply not fair. It is our job to teach – and teach well – not to facilitate. What happens when you have students work among themselves to figure things out: “A handful of students have some kind of idea what is going on, but with an eclectic mix of gaps in their knowledge and newly formed misconceptions. Some of these students are aware they have gaps and misconceptions, others are blissfully ignorant. And the rest of the students do not have a flipping clue what is going on. They are feeling confused and pretty down about themselves when they see their fellow classmates have figured it out. Many of those who failed to ‘discover’ the key relationships have already decided that indices are difficult, and yet another area of maths that they don’t understand. And so I wave goodbye to a group of confused students trundling out of the door, promising that we will pick this up again tomorrow, assuring them it will all be fine. I am already dreading the lesson, wanting to open up proceedings by saying ‘okay, everyone, forget what happened yesterday’. in the early knowledge acquisition phase, fully-guided, explicit instruction is the way forward. It is time efficient, minimises the chance of incomplete knowledge or the development of misconceptions, and via its positive impact on student achievement it is motivating. less guidance during instruction is simply not suitable for novice learners. More often than not, novices’ lack of domain-specific knowledge leads to a frustrating, demotivating experience. misconceptions never disappear from long-term memory, and the best we can hope to achieve is to make the correct knowledge more accessible. misconceptions are likely to lie close to the surface of our consciousness, fighting for our attention, and it can be rather effortful to avoid them in order to select the correct piece of knowledge. Successfully transferring knowledge from working memory to long-term memory is tricky enough, but in these cases we also need this new knowledge to battle against existing erroneous knowledge stored in long-term memory and come out victorious. If this existing knowledge is deeply embedded, then this battle may be an incredibly tough one, and simply telling students the right way of doing things as I suggested with adding fractions may not be enough. most of what we as teachers consider to be rote knowledge may in fact be inflexible knowledge, and the development of inflexible knowledge is a necessary step along the path to expertise. John von Neumann said: ‘In mathematics you don’t understand things. You just get used to them’. in mathematics you don’t understand things straight away. You just get used to them, and then you understand them. this – I just cannot keep quiet in lessons. I think it is due to some long-standing, misguided notion that silent classrooms are a bad thing, whereas noise is clearly a sign of learning. It took me 12 years, hundreds of pages of research, and a three-hour conversation with Dani Quinn on my podcast, but finally I now know that quite the opposite is true. When students are trying to concentrate, silence is golden. We need to give our students opportunity to go through the effortful process of attempting to retrieve information from their long-term memories, because it is this process of retrieval that leads to learning and retention. Working in groups: “explaining to others is potentially a dangerous thing to do during early knowledge acquisition phase” So much time is saved by cutting out the discussions and debates that used to infect my worked examples. Also, I find behaviour a lot less of an issue, because students know they must remain entirely silent – there are no grey areas. Engagement and activity were what I strived for precisely because they were things I could observe. Noise – in the form of discussion and questions – was good, whereas silence was a sign of disengaged, passive robots who simply could not be learning. discussions could easily last 10 or 15 minutes, and whilst confusion spread around the room, I could certainly see those students who understood becoming frustrated and even a little bit confused themselves. Students do the quizzes in silence, and on their own. The single most important thing is that these quizzes induce each student to retrieve information from their own long-term memories.

Sold a Story is just devastating. What a tragedy for so many students. I loved the last episode where the students were reading out loud. So delightful! I just finished a book, How I Wish I'd Taught Maths by Craig Barton. He's a teacher in the UK, so there's a fair bit of UK-specific stuff, but I have a couple of takeaways: Math teachers make a huge effort to make their classes "fun", and keep students "engaged", and whatever. Student engagement and fun is subjectively assessed by having students speaking to each other and debating and what not. This would have been my nightmare, and it's highly inefficient. Far more efficient and far more effective is to directly teach the students the stuff you want them to know. It's easier, faster, and you'll end up with students who are less likely to misunderstand things. I'm gobsmacked that teachers do it any other way. Math contests! OMG, it was only at nearly the last page when he mentions how helpful problems from the UKMT (bascially the UK equivalent of the AMC). It was so very gratifying to me, because this was exactly the way I was taught at my own high school. We had our regular math curriculum with the math teacher at the blackboard telling us how to solve problems. Then once a month we took a practice math contest and then we took the AHSME for reals. The perfect way to incorporate interleaving and spaced repetition and so very effective. I'm not familiar with "Envision." Do you have any photos or samples? But yeah, I agree with @Nichola, your multiplication and addition facts need to be reflex-fast or else pre-algebra is going to be painful. @Ellie I purchased Cuisinaire rods and then promptly gave them to another homeschooler after opening the package. Not much value there, IMO, but ppl seem to love them. Liping Ma's book is good, but my only takeaway is that it really helps if teachers have a solid understanding of math. Math teachers in China actually study elementary math. You can find the syllabi for the teacher training program at Stanford, and it completely lacks subject content.

@royspeed Those documents are beautiful! I'm now wishing I had little shadow boxes like on the right side of your transcript. I'm also wishing I had my course descriptions in the format you used.

You had me at "Pebbles." I wonder how grad student compensation vs. housing costs at UC compared with other comparable schools?

I worked through the introductory problems for my students, with me writing out the solutions on paper and describing what was happening as we went along. Then immediately following this introduction, my students complete the Exercises at the end of each section independently. Students check their answers in the Solutions Manual, and if they can't figure out what they did wrong they ask me. We set aside 1-3 days EACH on Review Problems and Chapter Problems at the end of each chapter; students also do these independently of me. The Review and Challenge Problems do a good job of checking for mastery. Also I like to see their AMC scores improve every year.

Control control control. No negotiation with school officials about what classes my kids can and can not take or whether they have fulfilled prerequisites. No relying on an overworked school counselor to write a letter. I can create my own beautiful transcript that is organized to present my student in the best possible light. No badly printed table with course name abbreviations that do not reflect the course content. Less teaching, more writing checks and finding options and opportunities.

You know I wonder, who are the kids who never watched Sesame Street or Electric Company? Those programs are full of phonics instruction, how did students miss that?

In writing a proof, whenever I make a statement, I ask myself, "How do I know this to be true?" If 2 segments appear to be the same length, and knowing they are the same is important to proving something, then I ask myself, "How do I know they are the same length?" Hint: it's usually congruent triangles, lol. But then I ask myself, "How do I know they are congruent?"

This is such a heartbreaking report. For more installments here are some more episodes that pre-date the Sold a Story compilation: At a Loss for Words Reading instruction in the US Reading instruction battles Reading curriculum is failing kids NY Times article I'm eager to hear the next episode of Sold a Story.

Caesar's English is lovely.

This question reminds me of the AoPS problem where you need to sort a bunch of numbers that have different exponents and different bases. So you end up comparing pairwise numbers that can be made to have either the same exponent or the same base and then putting them into order. All I can contribute is in terms of difficulty: regular book problems < starred book problems (usually, but not always)

Others on this board may be able to help you with these questions, but I also want to mention that the Common App support people are very responsive, and they are likely to give you fairly accurate information. (Not always a given, I know!) I've pinged Common App support and received a response within a couple of hours (maybe not on a weekend, I'm not sure). Good luck with these questions!

We did every problem in the textbook, and yes it takes longer, especially if your student is studying independently, but IMO it's time well spent. (We did not do alcumus.) Algebra is so foundational, you'll be glad you spent the extra time here. If however, you are in a rush, then you can skip the Challenge Problems. The problem sets are fairly short, but each individual problem is time-intensive, so when you start cutting out problems you are losing a lot of the value. Know, too, that a typical Algebra I class is about half of the AoPS Intro to Algebra textbook. It sounds like your student is off to a great start!

It has a title and an author, right? I put my textbooks in the course descriptions with just that information.

AFAIK, school counselors don't teach classes. They describe the student outside of class, more as person and less academically. The academic part is covered by his teacher recommendations. You've got this.

There are pts in gallon. How many gallons are in pts? The way I teach this is to start with: And then you can create different conversion factors: Divide both sides by to get and so Similarly, another conversion factor can be generated: Then I tell students that newbies will sometimes select the wrong conversion factor, generating something that looks like this: which does not answer the question, but it is technically correct. My question is how do we refer to these quantities with strange units, that aren't of use to anybody? Do we have a name for them?

I agree with you.

+1 for organizing team-type meet ups to prepare for First or maybe USACO. I'll bet a lot of parents won't really understand what a coding club or dojo would look like. I tried to organize a math club, and it didn't get off the ground. I had better luck organizing a MathCounts team, where I charged a nominal bit of money and I led the classes and got them prepared. Later my kids led the math teams. Still, it was always a struggle to recruit enough people to participate.

Have you thought about BFSU? You can scan the flow chart and see what your student is interested in learning. They teach the energies early in the series: kinetic/potential/etc. So it will help to have that taken care of first.

I keep thinking that operating a pop-up AP exam location would be lucrative.