What do we know about the science of teaching math?

[GoodnightMoogle]

I’ve been listening to the “Sold a Story” podcast, and as a former teacher I can’t say I’m surprised by the way money and hype drive curricula purchases over actual science. I cannot and will not ever understand why some people want to reject phonics, but that’s for another post.

We have a lot of good data now on how people learn to read. We aren’t machines and everyone is different, but for the most part, we do at least know that some strategies (like phonics) are useful and good, while others (like teaching kids to guess based on the picture or first letter) are less useful. 

Do we have any information like that out there for teaching math? The two big camps I see argue seem to be drilling memory facts vs conceptual knowledge. I would love to know if there is any good info out there on how we learn math and what really matters and helps us vs poor strategies.

All I can say is, when I taught in elementary schools, I was not a fan of Pearson’s “Envision” math. It jumped around a lot, tried to teach concepts, but had little drill. I had fifth graders counting on their fingers for problems like “6 + 5.” I don’t know what’s right, but something in my district was wrong. 

Edited December 22, 2022 by GoodnightMoogle

[Jean in Newcastle]

I am driving myself crazy trying to remember the title and author of a book about teaching mathematics that I found very influential.  I believe that the author was Chinese.  It dealt with the underlying concepts of math and it's progression and I think might have done some comparison with how things are handled in different countries. 

[Jean in Newcastle]

Found it;  https://www.amazon.com/Teaching-Elementary-Mathematics-Understanding-Fundamental/dp/0415873843

[Nichola]

I agree that the book by Liping Ma is really good. Another one I liked is The Teaching Gap by James Stigler and James Hiebert. They studied classrooms in the US, Germany, and Japan and discussed the differences among them. The part I remember best was that American teachers felt like they needed to make math fun by adding funny word problems or silly songs, while Japanese teachers thought that math was inherently enjoyable and that students would like math for itself.

In response to your question about memorization versus conceptual understanding, it seems like most books I’ve read talk about the importance of both. I’m not an expert by any means, but I have read a lot about math since we started homeschooling. I’ve read several times about how you can’t proceed to upper level math without memorizing math facts because people only have so much working memory. If you’re using all of it to figure out 9x8 because you don’t have it memorized, then you may struggle with the mental load of more difficult problems.  But it’s also hard to proceed to upper level math if you have only memorized algorithms and don’t have a flexible understanding of numbers and why the algorithms work, which comes from conceptual knowledge.

I’m actually reading a book right now that talks about this. Elementary Mathematics for Teachers by Thomas H. Parker and Scott J. Baldridge. Here’s a quote: “Ironically, the ability to recall these ‘basic facts’ is essential for the conceptual understanding of multiplication. It enables children to regard one-digit multiplication as trivial, which it is. That frees up short-term memory, allowing them to turn their attention to the overall structure of the problem. Conversely, children who have not memorized the basic facts will continue to think of multiplication as a procedure requiring time and attention. This can exhaust limited short-term memory. Children who do not know the basic facts are not merely slower at calculating, they have conceptual difficulties solving multi-step problems.” p. 29

 

[Ellie]

5 hours ago, GoodnightMoogle said:

Do we have any information like that out there for teaching math? The two big camps I see argue seem to be drilling memory facts vs conceptual knowledge. I would love to know if there is any good info out there on how we learn math and what really matters and helps us vs poor strategies.

I'm still fuzzy on what "conceptual knowledge" actually means. No one has 'splained that to me. Is it using manipulatives instead of traditional math (e.g., math that was last taught in the 50s)? That was the argument in the 80s when I started hsing, that *all* children had to use manipulatives (e.g., base 10 blocks, Cuisenaire rods) or they would be failures at math. Happily, I disregarded that information, because my dc and I would have perished if we'd had to do everything with some sort of manipulative.

[daijobu]

5 hours ago, GoodnightMoogle said:

I’ve been listening to the “Sold a Story” podcast, and as a former teacher I can’t say I’m surprised by the way money and hype drive curricula purchases over actual science. I cannot and will not ever understand why some people want to reject phonics, but that’s for another post.

Do we have any information like that out there for teaching math? The two big camps I see argue seem to be drilling memory facts vs conceptual knowledge. I would love to know if there is any good info out there on how we learn math and what really matters and helps us vs poor strategies.

All I can say is, when I taught in elementary schools, I was not a fan of Pearson’s “Envision” math. It jumped around a lot, tried to teach concepts, but had little drill. I had fifth graders counting on their fingers for problems like “6 + 5.” I don’t know what’s right, but something in my district was wrong. 

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.  

[Lucy the Valiant]

5 hours ago, Ellie said:

I'm still fuzzy on what "conceptual knowledge" actually means. No one has 'splained that to me. Is it using manipulatives instead of traditional math (e.g., math that was last taught in the 50s)? That was the argument in the 80s when I started hsing, that *all* children had to use manipulatives (e.g., base 10 blocks, Cuisenaire rods) or they would be failures at math. Happily, I disregarded that information, because my dc and I would have perished if we'd had to do everything with some sort of manipulative.

Example from my own childhood (so I'm not shaming anyone else): I had rock-solid applied-algorithm knowledge of mathematical concepts growing up. I got great grades, knew exactly what the teacher wanted me to write, and memorized ALL the formulas. Imagine my amusement when I realized . . . AS A HOMESCHOOLING MOTHER (with 2 degrees, mind you) . . . that the reason it's called a "square" number is that  . . . IT MAKES A SQUARE! 

Thankfully, I can chuckle. I was equally thrilled with cubes.

And you can bet your beautiful buttons that my own kiddos (while also rock-solid on formulas / facts / algorithms) have a much stronger CONCEPTUAL knowledge of the stuff they're doing in math. 🙂  

 

ETA: We had but didn't use much our actual manipulatives - cuisenaire rods, base 10's, etc. It was Singapore Math's bar models + Liping Ma that sort of woke me up and tuned me in, and honestly - once I actually SAW the mathematics, I was pretty much in love with them. ❤️ And couldn't get enough.

Edited December 23, 2022 by Lucy the Valiant

[Clarita]

25 minutes ago, Ellie said:

I'm still fuzzy on what "conceptual knowledge" actually means. No one has 'splained that to me. Is it using manipulatives instead of traditional math (e.g., math that was last taught in the 50s)? That was the argument in the 80s when I started hsing, that *all* children had to use manipulatives (e.g., base 10 blocks, Cuisenaire rods) or they would be failures at math. Happily, I disregarded that information, because my dc and I would have perished if we'd had to do everything with some sort of manipulative.

I think in order to figure out what is meant by "conceptual math" vs "traditional math" you have to take both to extremes. Most math curriculum that people like using is really some combination of the two or falls somewhere in the spectrum of the two. "Conceptual math" people picture "traditional math" as just rote memorization. You memorize the procedure for doing all the math things supplemented with memorizing all the "math facts", suppose that's multiplication table to 12?, I think addition and subtraction to 10?, I don't know if you memorize division tables (I mean if you have multiplication memorized you know the division). Then along those lines conceptual math teaches why all the procedures work, what we are actually doing to the objects and numbers in math. Taken to the extreme I guess then there is no memorizing because you just need to know how the whole thing works.

Hopefully in reality, the traditional math route at some point would teach why the long division procedure works, why carrying and borrowing works, and what exactly you are trying to do when you take the integral of something. Of course Liping Ma's argument is that in the US the whys behind a lot of the arithmetic procedures may never be taught to students because the math teachers themselves have not been taught. In reality, conceptual math also include a lot of practice and trying to get kids to do problems quickly so memorization happens naturally.

I've found manipulatives to be super helpful in teaching, even though my eldest is manipulatives adverse. I love using manipulatives to solve problems. It helps me to envision the problem I'm actually trying to solve and used them through Calculus and most definitely when I was working. However, for people who are generally good at math they don't need manipulatives for every lesson because a lot of the lessons are common sense for them. The people who are weak in math it can help to make math less abstract and daunting, especially those who tend to guess at math.

[daijobu]

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.
 

Edited December 23, 2022 by daijobu

[Clemsondana]

From my highly nonscientific anecdata from teaching my kids and volunteering at afterschool programs where I've worked with kids from many schools that use different approaches... both 'old fashioned algorithms and memorization' and 'singapore math style' programs can be effective.  What is commonly used is a hybrid that is terrible and has students drawing and counting for years, so they neither memorize arithmetic nor learn the reasoning skills to 'logic it out'.  Some things that I'd think would work don't.  One common thing here is that students are given a page with a grid of 5 days, 5 problems/day, for homework.  The problems are all sorts of different things - maybe a word problem, a long division problem, and equivalent fraction problem, etc.  They are supposed to help review.  But, in the absence of any book or example, a confused student has nowhere to start.  Finally, it is entirely possible to turn 'conceptual' approaches into bulky algorithms.   Here, to multiply 23 x 43, some students are taught to make a grid, like a punnet square, with 20 and 3 on the side and 40 and 3 on the top, and they multiply to fill in all 4 squares and then add them.  Most kids are good at this.  Then they do it with hundreds, with bigger grids.  In theory this teaches that you are actually breaking the numbers down, but in reality most kids have no clue what they are doing but find it easy to fill in the boxes.  This is going to be really tedious eventually.  Many kids don't actually have the conceptual understanding of what's happening, so when taught the traditional algorithm they think they've been taught something new that might give them different answers.  It's been quite an experience seeing how differently material is being taught.  

[GoodnightMoogle]

@daijobu thank you so much for this! Some of his observations actually remind me of the Sold a Story podcast. Basically, that teachers didn’t like some of the traditional phonics instruction because “it seemed dull.” Kids facing the teacher, lots of direct instruction. I think teachers have fooled themselves into thinking that kind of learning isn’t good enough, for some reason. We think we need flash and pomp and excitement. Ugh. And the worship of student on student discussion. Like it has its place but I don’t understand why we are so allergic to … teaching? I think that’s why math gets so drawn out in the classroom setting, as well. All these extras added on, and suddenly math is an hour and half long for second graders. 

[daijobu]

2 hours ago, GoodnightMoogle said:

@daijobu thank you so much for this! Some of his observations actually remind me of the Sold a Story podcast. Basically, that teachers didn’t like some of the traditional phonics instruction because “it seemed dull.” Kids facing the teacher, lots of direct instruction. I think teachers have fooled themselves into thinking that kind of learning isn’t good enough, for some reason. We think we need flash and pomp and excitement. Ugh. And the worship of student on student discussion. Like it has its place but I don’t understand why we are so allergic to … teaching? I think that’s why math gets so drawn out in the classroom setting, as well. All these extras added on, and suddenly math is an hour and half long for second graders. 

@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.   

Edited December 23, 2022 by daijobu

[Ellie]

Huh. Well, I'm still not much more enlightened than I was before, lol. 🙂

[GoodnightMoogle]

9 minutes ago, Ellie said:

Huh. Well, I'm still not much more enlightened than I was before, lol. 🙂

For me, I think I had great procedural knowledge taught to me. Lots of speed drills and flashcard games. I knew my math facts! But I lacked any deeper understanding .

For example: I made it all the way to AP Calculus. I have a great memory for formulas! And I could recognize patterns enough to read a word problem and know which formula I was supposed to use. I did great in the class; I can still see those different curved graphs I had to make. I never understood what they meant though! I had no idea what I was *actually* doing. I think that’s what learning concepts is. Knowing *why* that formula works and what that graph actually means. 

On the bright side, I may not know anything about math, but I *can* still sing the quadratic formula out in my head!

 

 

 

[Not_a_Number]

I’m really, really interested in this, but what I know is that there aren’t good studies on this. And a lot depends on a student’s ability to connect the dots, too.

In my opinion, an approach that’s largely algorithmic will on average produce students without conceptual understanding. Of course, as @Clemsondanasays, lots of “conceptual” approaches wind up being learned simply as more tedious algorithms.

I also tend to think we undervalue the role of communication in math. To me, math is a great way to teach kids to reason and to use linear logic. I don’t tend to see an emphasis on that in either conceptual or algorithmic approach.

[Xahm]

This also interests me a lot, so I'm so glad it's a topic here! I'd love to see real evidence, but it's hard to get. I was looking at some elementary math homework a few years ago to help a kid, and it had instructions to the parents for how to help their kids. They were supposed to read through word problems for key words that told them which operation to use, and it listed the key words. I don't remember all of them, but they were told "share" means "divide." That sounds reasonable until you have "Eleanor had 8 cookies until she shared some with Paul. Note she has 4. How many did she share?" That would be easy if you think through the story of the problem, but the key word method gives the answer "2." I have a feeling, though, that the program just avoids ever giving them problems that don't work with their algorithm. If the same publisher also makes the standardized tests, those kids are going to look far more proficient than they probably are, making it hard to judge the merits if the program.

[FreyaO]

19 hours ago, GoodnightMoogle said:

For me, I think I had great procedural knowledge taught to me. Lots of speed drills and flashcard games. I knew my math facts! But I lacked any deeper understanding .

For example: I made it all the way to AP Calculus. I have a great memory for formulas! And I could recognize patterns enough to read a word problem and know which formula I was supposed to use. I did great in the class; I can still see those different curved graphs I had to make. I never understood what they meant though! I had no idea what I was *actually* doing. I think that’s what learning concepts is. Knowing *why* that formula works and what that graph actually means. 

On the bright side, I may not know anything about math, but I *can* still sing the quadratic formula out in my head!

 

 

 

I think part of the issue is the compartmentalization of knowledge in school. There are obvious and beautiful illustrations of all these concepts in science. But I am told that Gen Ed Physics in US HS is "qualitative". 

[Clarita]

3 hours ago, FreyaO said:

But I am told that Gen Ed Physics in US HS is "qualitative". 

Well Gen Ed Physics in US High School does not require Calculus. I think a typical college bound US High schooler gets to Calculus sometime their senior year. Maybe other countries get to Calculus sooner?

4 hours ago, Xahm said:

They were supposed to read through word problems for key words that told them which operation to use, and it listed the key words.

I was for sure taught that method in elementary school. I believe Singapore Math and/or Beast Academy says don't do that. Instead have the students actually think through what the problem is and provide manipulatives if necessary.

[LauraClark]

On 12/22/2022 at 7:08 PM, Lucy the Valiant said:

Example from my own childhood (so I'm not shaming anyone else): I had rock-solid applied-algorithm knowledge of mathematical concepts growing up. I got great grades, knew exactly what the teacher wanted me to write, and memorized ALL the formulas. Imagine my amusement when I realized . . . AS A HOMESCHOOLING MOTHER (with 2 degrees, mind you) . . . that the reason it's called a "square" number is that  . . . IT MAKES A SQUARE! 

Thankfully, I can chuckle. I was equally thrilled with cubes.

And you can bet your beautiful buttons that my own kiddos (while also rock-solid on formulas / facts / algorithms) have a much stronger CONCEPTUAL knowledge of the stuff they're doing in math. 🙂  

 

ETA: We had but didn't use much our actual manipulatives - cuisenaire rods, base 10's, etc. It was Singapore Math's bar models + Liping Ma that sort of woke me up and tuned me in, and honestly - once I actually SAW the mathematics, I was pretty much in love with them. ❤️ And couldn't get enough.

Similar experience for me-I did great in math and loved it through school, but now that I'm teaching it there are a lot of lightbulb moments for me. However, based on that, I've gone a bit in the other direction with teaching than you have (not that you're wrong, just another perspective): I'm not so worried that my kids don't understand the why's behind the facts they learn-they will understand the why's as they get older (and I do plan on making sure they understand them as they get older). I tried early on to explain the logic behind the ones and tens and hundreds place, for example, but it fell on tiny deaf ears, so I just had them memorize "ones, tens, hundreds" so they could make the place accurately.

I think the classical model as a whole encourages this: the grammar age is not about the why's, but more about memorizing and getting the information in their heads. As they progress through the stages the why's should start to be understood. If you think about it, that's really what reading/phonics is all about too: I have learned a lot as an adult teaching phonics-things that I just memorized as a child but now understand the reasoning behind.

[Clarita]

I think it's also important to remember for all the science that may or may not be out there, when science says is the "best" may not be the best for a particular child. Sometimes it's just statistically the best meaning if we teach 100 students using a particular method 95 come out doing better, if you are talking about teaching one child you may have 1 of the 5 children who may be served better by another method or is just going to struggle no matter what. Or conversely the 1 of the 5 who could just learn the material through osmosis. 

8 hours ago, LauraClark said:

I'm not so worried that my kids don't understand the why's behind the facts they learn-they will understand the why's as they get older (and I do plan on making sure they understand them as they get older). I tried early on to explain the logic behind the ones and tens and hundreds place, for example, but it fell on tiny deaf ears, so I just had them memorize "ones, tens, hundreds" so they could make the place accurately.

My eldest on the other hand won't learn material unless he knows the why. He could only recognize 5 letters until he learned letters could symbolize sounds and could be used to decipher the words on the page. (I tried with all the letter immersive experiences.) The next day (I kid you not) he started reading, even words with the 21 letters he couldn't recognize the day before.

[UHP]

On 12/22/2022 at 6:58 PM, Ellie said:

I'm still fuzzy on what "conceptual knowledge" actually means.

I've grown allergic to the phrase. Some reformers complain that math education is too focused on "procedures" and not enough on "concepts." But my experience so far is that no one who makes this critique can do all three of the following:

1. Assemble a long list of concepts that they would like kids to know.

2. Explain how to tell apart the kids who do and the kids who don't know each concept on the list.

3. Describe teaching procedures that will instill knowledge of the concept in the kid.

Common core at least did 1, but their list is shabby. There are "common core exams" but they are designed to give feedback to states and schools, not students. I don't regard them as a good answer to 2. And if they had a theory about 3, I think the evidence is that it hasn't worked.

On 12/22/2022 at 7:52 PM, daijobu said:

notes and highlights from the Barton book. 

Thank you for posting these!

[Clarita]

I am on the fence about how much math "every" child should know. I know my math goals for my own children, but I don't think all kids should or could get to that point. Realistically I don't even know if my kids (6 and 4.5) would meet that goal because I want them to know and understand more math than I do and I did calculus through college.

6 hours ago, UHP said:

2. Explain how to tell apart the kids who do and the kids who don't know each concept on the list.

I just know tutoring in high school and occasionally tutoring some community college students or helping out a friend, a lot of people who "aren't good at math" usually have a topic or concept that they just didn't get. I see a lot of the struggle come when moving from arithmetic to algebra. Some of them struggle when they are exposed to steps done in a different order in algebra and it really throws them off and they start guessing or there are too many different steps in algebra and they don't have the capacity to remember all the steps. Those kids though sometimes get categorized as not having learned procedures well either. 

The next group of kids do fine throughout math. They just might not be able to perform the tricks some people do by computing things they don't already know super quickly. There is a tiny stumbling block when they get into a job where they have to find solutions to problems that haven't been solved yet. So if all they know is the procedure but not the underlying principles behind what's going on they have a bit of a steep learning curve when they get to work and suddenly no one knows the procedure because well no one has done this before. There are plenty of jobs in the world that do not require these skills. 

I'm not trying to reform education for the US though. Just a mom who teaches better what is deemed more "conceptual" curriculums and whose kids seem to do better with conceptual teaching. I thoroughly understand my kids are super young and are genetically pre-disposed to be good at math.

 

 

 

[UHP]

28 minutes ago, Clarita said:

I'm not trying to reform education for the US though. Just a mom who teaches better what is deemed more "conceptual" curriculums and whose kids seem to do better with conceptual teaching.

Are you referencing any specific "conceptual curriculums" here? What have you been doing with your 4- and 6-year-old? (Whatever you're doing I certainly didn't mean to criticize it, or you, for using the word "conceptual")

[Jean in Newcastle]

I had many math teachers who taught nothing more than algorithms. When I would ask “why” we did things a certain way the answer was “just do it”. I’m quite sure that it’s because they didn’t actually know the why. I actually didn’t learn some of those concepts behind the algorithms until I was in a “math for teachers “ class. This might not be true for everyone but I did so much better in math and actually liked math better when I had the concepts as “hooks “. I don’t remember rote things very well unless they make sense to me. 
 

I am very glad that I have been able to teach all my students (including my own children) the concepts behind the algorithms. A calculator can run rote algorithms. A true mathematician thinks mathematically, which means conceptually. 

Edited December 27, 2022 by Jean in Newcastle
Typo

[Clarita]

11 hours ago, UHP said:

Are you referencing any specific "conceptual curriculums" here? What have you been doing with your 4- and 6-year-old? (Whatever you're doing I certainly didn't mean to criticize it, or you, for using the word "conceptual")

4 year old uses Kindergarten with Confidence, but at the kindergarten level probably all math curriculum is pretty conceptual. 

6 year old started with Singapore Math and did Kindergarten and part of level 1 but when Beast Academy 1 came out moved to using Beast Academy. 

DH and I use a lot of math in daily life both normal everyday usage and doing calculations in our heads, or just for fun discussions. Sometimes the for fun discussions is on a math topic that the kids aren't ready to learn algorithmically or actually how to solve so they get a loose conceptual description from us. Then we also find math tricks super interesting so we talk about those and then we like to find out why and how the tricks work and when they are applicable, etc. we don't do it for the kids' benefit but if they happen to be present then they will be included. So it's nice to have curriculum that weave in this figuring out and breaking apart numbers, because then they are more clued in during the conversation.

[BusyMom5]

I have my own opinions regarding math.  I feel like the concept camp forgets how important it is to build fluency.  The algorithms camp forgets that kids need to understand the concept,  too.  Both are important.   What both camps forget are that kids are not repositories of all information inputted.  They forget.  They don't listen.  They jump to conclusions and find little math tricks.  Even posters who claim they weren't taught a concept- its possible you were, but you latched onto the algorithm and just never put the full concept into memory. 

When teaching my own kids I often hear "I can just put it into my calculator like this!" (Inputs long equation) = Correct Answer.  My boys are in Algebra 2 and I hear this all the time.  They often jump to conclusions about how to solve and kinda tune me out- even when I know they aren't learning what they are supposed to!  They try to logic their way through,  and often it will work but be clunky.  We've recently been doing completing the square.   I taught it- they weren't paying attention.  Ive since taught it to them separately several times.  One has memorized how to do it and does not care to know the logic.  The other one- he gets thrown by why that works, even though I've worked it forwards and backwards- he needs to get the concept.  I cannot imagine doing this level of math with kids who fo not have a solid base of math facts memorized.  

One of mine really struggled with math.  I tried to teach the logic, but sometimes it was just beyond her and she hates math.   I settled for memorizing algorithms.   It varied based on the topic.  She would either get it.... or it was a foreign language.  There was no in-between.  She got through College Algebra with a B and has kissed math goodbye.  Shecwill have to take another stats class, but she says that's different.  

So- all of that to say that no matter how one teaches, success depends on the student. 

[Jean in Newcastle]

Saying that it’s “algorithm vs concepts” is a false dichotomy. Of course you need both. 

[UHP]

2 hours ago, Jean in Newcastle said:

Saying that it’s “algorithm vs concepts” is a false dichotomy. Of course you need both. 

The pupil needs to learn both algorithms and concepts, but there is an important asymmetry from the point of view of the tutor. Students accumulate misunderstandings about algorithms and concepts all the time. The teacher has to pre-empt these misunderstandings if possible and remedy them when not possible. To do this effectively, the teacher has to be constantly probing what the kid knows or thinks he knows.

Now, how can a teacher tell if what a student knows? You can't reason that, just because you have explained it well, or explained it multiple times, that the student has understood. You have to test them. It is easy to test if they know an algorithm: you just have to watch them execute the algorithm on a range of examples. It is hard to test if they understand a concept. To do it well requires an extremely detailed analysis of the concept, attending to details that are hard for adults who've already mastered the concept to see.

My sense (or anyway my complaint) about many professionals who write about "conceptual approaches to teaching math" is that they don't address this difficulty at all. They often are very excited to discuss different creative ideas for teaching a concept, and have nothing to say about how to figure out if a kid has learned the concept or not.

[Jean in Newcastle]

2 hours ago, UHP said:

The pupil needs to learn both algorithms and concepts, but there is an important asymmetry from the point of view of the tutor. Students accumulate misunderstandings about algorithms and concepts all the time. The teacher has to pre-empt these misunderstandings if possible and remedy them when not possible. To do this effectively, the teacher has to be constantly probing what the kid knows or thinks he knows.

Now, how can a teacher tell if what a student knows? You can't reason that, just because you have explained it well, or explained it multiple times, that the student has understood. You have to test them. It is easy to test if they know an algorithm: you just have to watch them execute the algorithm on a range of examples. It is hard to test if they understand a concept. To do it well requires an extremely detailed analysis of the concept, attending to details that are hard for adults who've already mastered the concept to see.

My sense (or anyway my complaint) about many professionals who write about "conceptual approaches to teaching math" is that they don't address this difficulty at all. They often are very excited to discuss different creative ideas for teaching a concept, and have nothing to say about how to figure out if a kid has learned the concept or not.

Nonsense. All you have to do is to ask the student to explain the concept. If you actually understand it yourself, it’s not that hard. 

[wendyroo]

4 hours ago, UHP said:

Now, how can a teacher tell if what a student knows? You can't reason that, just because you have explained it well, or explained it multiple times, that the student has understood. You have to test them. It is easy to test if they know an algorithm: you just have to watch them execute the algorithm on a range of examples. It is hard to test if they understand a concept. To do it well requires an extremely detailed analysis of the concept, attending to details that are hard for adults who've already mastered the concept to see.

For the 3rd to 5th grade students I tend to tutor in math, I find it takes very few questions to ascertain their level of conceptual understanding. My goal is to find questions that with conceptual understanding are trivial, and without conceptual understanding are convoluted.

A typical question would be 5/7 + ? = 1

Or, Figure out in your head how many 8s fit into 816?

If they can solve those in a matter of seconds, then they have some level of conceptual understanding of fractions, place value and division. Especially for the second question, speed and doing it mentally really matter - I want to know if they "see" how place value makes it a trivial question or if they have to use the algorithm to work it out.

[UHP]

4 hours ago, Jean in Newcastle said:

Nonsense. All you have to do is to ask the student to explain the concept. If you actually understand it yourself, it’s not that hard. 

If they give a beautiful and articulate explanation when you ask, then of course you know your work is done. If they give the kind of answer my 8-year-old is more likely to give, it is quite a bit more ambiguous. My experience is that the capacity for understanding comes far in advance of the capacity for explaining; that goes for kids and adults both. Of course I couldn't prove it to you by your method.

1 hour ago, wendyroo said:

A typical question would be 5/7 + ? = 1

Or, Figure out in your head how many 8s fit into 816?

I like these examples, agree that a kid who answers them correctly and quickly is likely to understand something well. But I think you still have a lot of digging to do if a kid gives a wrong answer, or a slow one.

Side comment on the fractions question: I think 9/7 - ? = 1 or 9/7 + ? = 3 would give more information, specifically because many kids carry around the unfortunate idea that "a fraction is something that is less than one."

[Jean in Newcastle]

12 minutes ago, UHP said:

If they give a beautiful and articulate explanation when you ask, then of course you know your work is done. If they give the kind of answer my 8-year-old is more likely to give, it is quite a bit more ambiguous. My experience is that the capacity for understanding comes far in advance of the capacity for explaining; that goes for kids and adults both. Of course I couldn't prove it to you by your method.

I like these examples, agree that a kid who answers them correctly and quickly is likely to understand something well. But I think you still have a lot of digging to do if a kid gives a wrong answer, or a slow one.

Side comment on the fractions question: I think 9/7 - ? = 1 or 9/7 + ? = 3 would give more information, specifically because many kids carry around the unfortunate idea that "a fraction is something that is less than one."

Explanations can be in the form of questions like Wendy posited. I wasn’t really expecting children to give conceptual “proofs”. Anyway, the point is that conceptual understanding is not some big unknowable thing. 
 

Another way that I can pinpoint certain types of conceptual holes in my student’s understanding is to see what kind of mistakes they make. While some mistakes are obviously computational, some show a lack of understanding of the concepts that are involved. 

[wendyroo]

1 minute ago, Jean in Newcastle said:

Explanations can be in the form of questions like Wendy posited. I wasn’t really expecting children to give conceptual “proofs”. Anyway, the point is that conceptual understanding is not some big unknowable thing. 
 

Another way that I can pinpoint certain types of conceptual holes in my student’s understanding is to see what kind of mistakes they make. While some mistakes are obviously computational, some show a lack of understanding of the concepts that are involved. 

This is the second prong to my approach.

If I meet a tutoring student and probe a concept like fractions with a few questions similar to what I posted, then...

...if they get them right, I start tutoring at about grade level and scrutinize their mistakes (because clearly they are making plenty or they wouldn't be in tutoring) to determine their weaknesses.

...if they get them wrong, I start tutoring from the ground up, literally going back to dividing shapes into equal size parts and naming the fractions, because they are clearly missing fundamental concepts and do not currently have a strong enough foundation upon which to build grade level skills. Obviously I still watch their mistakes carefully, but in this case I spend far less time trying to pinpoint their holes, and far more time just trying to lay a completely new framework.

[Clarita]

In my humble opinion, more than conceptual vs. algorithmic, the thing I would say holds students back today is the normalization that math is hard and it's OK not to know. 

[Not_a_Number]

I don’t think there’s such a thing as “getting” a concept. Rather, there are levels of fluency, like with reading. 

When we say a kid can read, there are lots of questions about what that means. Are they sounding things out? Reading haltingly? Reading out loud at a normal pace? Reading quickly in their heads?

Most concepts are the same way. It takes repeated (and non-algorithmic) exposure to be really fluent with a concept. In my opinion, many people don’t reckon with that enough and often jump to algorithms/memorization before kids are ready.

Edited December 31, 2022 by Not_a_Number

[UHP]

7 hours ago, Not_a_Number said:

I don’t think there’s such a thing as “getting” a concept. Rather, there are levels of fluency, like with reading. 

For math, I don't agree. Perhaps because I take a microscopic view of what is a "concept."

"Negative numbers" are not a microscopic concept. It could take a long time to become fluent with negative numbers.

But, "absolute value of a signed number" is a microscopic concept. It takes only a few minutes for a kid to become fluent with it, and you can test them by presenting just a few problems. (Actually I would divide it into two concepts: "absolute value without the notation" and "absolute value with the notation." The problems "without the notation" have the form: which of +4,+8 is farther from zero? which of +1,-8 is farther from zero? etc. The problems "with the notation" have the form: figure out |-4|, |+9|, etc..)

My view: the large "negative numbers" concept is made up out of dozens (at most) of microscopic concepts. A kid can be made fluent with negative numbers very efficiently by identifying these constituent concepts and teaching them explicitly one at a time, to mastery.

Language isn't logical, math is logical. Because of this, there are a thousand times as many rules or more that have to be mastered to be fluent in a language as to be fluent in math. Infinitely many of the rules contradict each other, like how to pronounce words that end in "ough." It takes a long time to get used to this network of contradicting rules.

The rules in math contradict each other extremely rarely, and usually in very anodyne ways. (An example of this very rare anodyne thing, that I watched out for early on tutoring my 5-at-the-time-year-old: sometimes "x" means times and sometimes it stands for a variable). Instead of contradicting each other, they reinforce each other: mastering one concept is an aid to mastering many others.

"Micro" suggests "millionth" but I think there are not millions but only a few thousand of these microconcepts that come up in K-8 education, and, especially since they are mutually reinforcing, that they could be taught over the course of just a couple of years to a large number of children. I might even say to most children.

[Jean in Newcastle]

Kids do not learn and retain mathematical concepts in a linear fashion. They can understand it initially but it takes time for it to become part of their thinking. That process is indeed layered and takes repeated exposure for fluency. Just as Numbers says. 

[8filltheheart]

9 minutes ago, Jean in Newcastle said:

Kids do not learn and retain mathematical concepts in a linear fashion. They can understand it initially but it takes time for it to become part of their thinking. That process is indeed layered and takes repeated exposure for fluency. Just as Numbers says. 

I agree. This conversation is the same conversation as the critical thinking one. Higher order thinking skills are reliant on lower order skills. Higher order thinking skills intertwine the lower order ones if a child is processing learning correctly.

Bloom knew what he was talking about in general. Learning is not completely linear, but kids can't analyze or evaluate what they have no knowledge or understanding of.

I personally see algorithm/conceptual as a false dichotomy. They work together. 

[UHP]

21 minutes ago, Jean in Newcastle said:

Kids do not learn and retain mathematical concepts in a linear fashion.

I don't know what you mean. Everyone experiences life in a linear fashion; that includes their education. When presenting information to kids (or to anyone), you have to choose what order to present it in. Such choices are very consequential: ordering them in the right way can save an enormous amount of time in our brief and linear lives.

[wendyroo]

12 minutes ago, Jean in Newcastle said:

Kids do not learn and retain mathematical concepts in a linear fashion. They can understand it initially but it takes time for it to become part of their thinking. That process is indeed layered and takes repeated exposure for fluency. Just as Numbers says. 

I agree.

When I teach something like absolute value (even a "microscopic skill" such as absolute value with notation), I expect it to take many exposures. And I don't expect true retention or understanding until the child has seen it enough times, in enough contexts, that they can start fitting it into their established mental framework and linking it to other fully mastered skills.

The topic of negative numbers came up with my 7 year old the other day when we were talking about the temperature outside. This was certainly not her first exposure to negative numbers, so I decided to play around with addition and subtraction a little bit ie If it is -2 degrees, how much would it have to warm up for it to be 0 degrees? and questions like that. That was easy for her, so we progressed to "The difference between -3 and 5 degrees" and problems of that nature. She did well with that concept, so as we were winding down, I threw out the term absolute value. I explained that absolute value is the distance from a number to 0, so to find the difference between -3 and 5 we were actually thinking about |-3| + |5|. I linked that to the concept of elapsed time which we had just been working on: 7am to 2pm = 7am to noon + noon to 2pm.

We were doing this whole lesson mentally while looking at a number line, so I briefly introduced absolute value notation by explaining about the two bars, and then we used our fore arms raised on either side of our head as the absolute value notation as we said various numbers. There was a lot of giggling. We played around just long enough for her to realize the conundrum that if I pointed to -7, that she could easily name that |7|, but that if I called out |7|, that she could not definitively know where to point on the number line.

I certainly don't expect full retention or mastery of "absolute value with notation" yet. We will spiral back to this many time, in different ways, going deeper and deeper - just like we do with all concepts.

[wendyroo]

14 minutes ago, UHP said:

I don't know what you mean. Everyone experiences life in a linear fashion; that includes their education. When presenting information to kids (or to anyone), you have to choose what order to present it in. Such choices are very consequential: ordering them in the right way can save an enormous amount of time in our brief and linear lives.

I can certainly teach in a linear fashion (though I often don't choose to do so dogmatically), but my children rarely learn in a linear fashion.

My 7 year old can add $247.84 + $399.67. She can figure out that in our state she would have to pay 51 cents in tax on a $8.50 purchase. She can think of debt as negative money. She can figure change on purchases.

What she can't do it remember which coins are which and how much they are worth. She just can't.

Certainly adding coins is a much lower level skill in the linear progression of currency math. Every curriculum is going to teach adding two dimes before they tackle adding decimal currency or calculating tax. And, by golly, I have been trying to teach her the coins for years now. But I don't have control over what she learns, and her skills and conceptual understanding are far from linear (along any axis, academic or otherwise).

[Clarita]

25 minutes ago, UHP said:

When presenting information to kids (or to anyone), you have to choose what order to present it in. Such choices are very consequential: ordering them in the right way can save an enormous amount of time in our brief and linear lives.

I don't always get to choose when education topics happen. Sometimes they just happen because the things we learn are things we need to use for life. I certainly hope my botched explanation of investing to my 6 year old last week is not consequential. 

[UHP]

7 minutes ago, wendyroo said:

My 7 year old can add $247.84 + $399.67. She can figure out that in our state she would have to pay 51 cents in tax on a $8.50 purchase. She can think of debt as negative money. She can figure change on purchases.

What she can't do it remember which coins are which and how much they are worth. She just can't.

My 8-year-old has the same skills and the same gap. In our case, it's because she and many of her friends have hardly any experience of money. Even at the grocery store she watches me pay with a credit card instead of counting out bills and coins. It was different when I was a kid. I taught her to figure the tax and to compute change not "out in the field" but in our math lessons.

But you aren't describing anything that's not linear. You have only proved by example that knowing how many cents in a nickel and how many nickels in a dollar are not prerequisite skills for working with decimals, percentages, and tax. And why should know it be ? When you examine those concepts (decimals, percentages...), even in a rather cursory way, you can see that they have nothing to do with how many cents in a nickel.

[UHP]

23 minutes ago, Clarita said:

I don't always get to choose when education topics happen.

Me neither.

[wendyroo]

19 minutes ago, UHP said:

My 8-year-old has the same skills and the same gap. In our case, it's because she and many of her friends have hardly any experience of money. Even at the grocery store she watches me pay with a credit card instead of counting out bills and coins. It was different when I was a kid. I taught her to figure the tax and to compute change not "out in the field" but in our math lessons.

But you aren't describing anything that's not linear. You have only proved by example that knowing how many cents in a nickel and how many nickels in a dollar are not prerequisite skills for working with decimals, percentages, and tax. And why should know it be ? When you examine those concepts (decimals, percentages...), even in a rather cursory way, you can see that they have nothing to do with how many cents in a nickel.

Then I guess I'm not clear on what you mean by linear progression.

Obviously, kids can't learn 4 digit addition with regrouping until they have learned 3 digit addition with regrouping. But I would really count those as the same skill (despite many curriculum holding out for a long time before teaching the 4 digit version like it is a much more difficult skill), and argue that if a child has truly mastered 3 digit addition with regrouping then they are fully capable of any number of digits.

I guess I don't see a lot of strict prerequisites in arithmetic. I do more so in upper level math - it is hard to factor polynomials if you don't understand exponents (though I think a large percentage of students try to do just that by rotely following algorithms) - but elementary math feels more to me like a web of complexly interconnected skills rather than neat skill lists that need to be learned in order.

[Jean in Newcastle]

Not all mathematical education is linear. A lot of it is spiral. Especially in countries outside of the US.   That doesn’t mean that there are a progression of skills but not everything falls into a neat little line. 

[maize]

I experienced what happens when there are significant gaps in math progression.

Between ages nine and sixteen I moved internationally four times. I lived in five different countries on three different continents, and attended schools in three different primary languages of instruction --two of them new to me when I was enrolled in the school. Both because of language barriers (I wasn't learning much content in class until I picked up enough of the language to follow along) and because of significant variations in scope and sequence from country to country and school to school, I missed out on a lot of important instruction. That really didn't impact me significantly in any subject but math, but by my last couple of years of high school I was really struggling with math. Among other things, I missed out completely on large portions of what is taught in a standard US progression in pre-algebra and algebra 1. 

I'm pretty good in general at conceptualizing math, but I couldn't fill in all the gaps on my own.

Edited January 1, 2023 by maize

[UHP]

2 hours ago, wendyroo said:

Then I guess I'm not clear on what you mean by linear progression.

The phrase that I used, following Jean, was "linear fashion." I wouldn't use "linear progression" because to my ear it suggests that the kids don't forget and don't ever need remedies.

By "linear fashion" I don't mean anything cryptic or metaphorical: only that kids learn in a linear fashion because they experience time in a linear fashion. In time, our lives are a straight line — the obvious fact has nonobvious consequences for education, or at least consequences that many educators have a blind spot about. The main consequence for a tutor is this: if there is a set of things you would like your pupil know, you have to decide what order to present them in. My claim is that the order matters a lot.

When you are talking about very large concepts, you have few options for how to order them. Should you teach negative numbers before fractions? I don't have a strong opinion. But when you zoom in on what I am calling "microscopic concepts" you have many many more options for how to order them, and among those options are opportunities for great clarity, efficiency, and acceleration.

For example, one of the common core standards for kindergarten is that children are to learn to write numbers from 0 to 20, and in grade one they are to learn to write the rest of the two-digit numbers. (Actually the standard 1.NBT.A.1 is to read and write numerals less than 120). These are both microscopic concepts or microscopic skills: I believe they are presented in the wrong order. I believe it is far more expedient to teach kids (in kindergarten or before) first to read and write two-digit numbers that are not in the teens, and then to treat the teens only when they have mastered the larger numbers. It would save a lot of kids a lot of grief. ("Numbers that start with 1 are tricky. We don't say 'ten six' and we don't say 'tenty six' or "onety six." We say 'sixteen.'")

To be concrete about something more advanced, here are the microscopic skills and concepts that I think make up "expertise with negative numbers," at the sixth or fifth grade level. This is cribbed from the Teacher's Guide to "Connecting Math Concepts Level F," which I adore. A weakness of this list (unusual for Connecting Math Concepts) is that it does not indicate how signed numbers can be incorporated into word problems.

1. Being able to combine the values in a problem that adds and subtracts more than one value: (Look at 0+26-15+104-2-8, figure out the total added, figure out the total subtracted, then work the simple subtraction problem)

2. Being able to rewrite and work a problem with signed terms in a different order: (Look at the problem 10 - 2 - 3 = ?. Rewrite it in the new order 2, 3, 10. The correct answer is -2 -3 + 10 = ?.)

3. Being able to locate signed numbers on a number line. (Just being able to put their finger on -5 and +7 when you ask them to.)

4. (skipping some more number line skills that are hard to explain in this format)

5. Being able to indicate which of two signed numbers has greater absolute value ("is farther from zero").

6. Being able to figure out the magnitude of the answer to a column problem with signed numbers. (If the signs are the same, you add the absolute values, if the signs are different, you subtract the absolute values.)

7. Being able to figure out the sign of the answer to a column problem with two signed numbers. (It's the sign of the one that has larger absolute value)

8. Combining skills 6 and 7

9. (If they know the distributive property already) Apply the distributive property to expressions or column problems like (-15+8-2+3) x 5. 

10.  Apply the distributive property to expressions or column problems that multiply by a negative number, like (-15+8-2+3) x (-5). (This is a nice way to introduce the sign-flipping rule)

11. Being able to work signed number problems of the form (-4) x (-3). 

12. Being able to divide by signed values.

13. Being able to work signed number problems of that add and subtract multiply signed numbers, like (-4) - (-3) + 7 +(-16).

If a kid has mastered the prerequisites, she can learn one skill a day in a five-minute session, and practice the previous skills she's learned over another five minutes, and thereby obtain "expertise with negative numbers" over the course of a little more than two hours spread out over a little less than three school weeks. (Of course if she doesn't put that expertise to use right away, she is likely to forget... this is part of my objection to the "spiral curriculum" that I might get on a high horse about later.)

Edited January 1, 2023 by UHP

[UHP]

42 minutes ago, maize said:

I'm pretty good in general at conceptualizing math, but I couldn't fill in all the gaps on my own.

Did you make much math progress as an adult, or does this still trouble you?

[Clarita]

41 minutes ago, UHP said:

I believe it is far more expedient to teach kids (in kindergarten or before) first to read and write two-digit numbers that are not in the teens, and then to treat the teens only when they have mastered the larger numbers. It would save a lot of kids a lot of grief. ("Numbers that start with 1 are tricky. We don't say 'ten six' and we don't say 'tenty six' or "onety six." We say 'sixteen.'")

There is a study on this (back to the topic) (https://www.wsj.com/articles/the-best-language-for-math-1410304008 references this) where a person's spoken language affects early math skills. Some languages are easier to count in than others. I like Singapore Math's method of just allowing kids to say one ten and six more. Then kids have a strong base in place value and all the numbers. Some time later they can remember sixteen vs sixty-one or fifty vs five-ty.