The video. I don’t know what the situation is in this particular school - you say they get good results - but I would not recommend doing this at all.
This class previously spent a week doing similar activities to work out things like ¾ x 6=8. Now they are spending a long painful class trying to figure out ¾ x 9 in the same way. Why? What next? Another week working out 2/17 x 5? Yes, I know this is supposed to be “understanding.” No, it’s just a complicated way of working out a problem using pictures.
You can demonstrate some simple examples for students using pictures to get the point across, but this doesn’t need to go on for so long. We should be trying to make things less complicated for students, not more complicated.
Some points:
a) This approach is confusing and not all students will be able to follow this. Even in this class, not all students are participating. Note the one boy (~2:30) who answers something incorrectly (the teacher doesn’t correct him?) and then puts his head down. I highly doubt this is going to work out well in your average class of 5th graders. Some students get to be the stars that come up with all the ideas; other students just feel stupid.
b) This is incredibly time consuming. If there were huge benefits to this, that may be reasonable. I’m not convinced. Students would be better off spending more time practicing fraction arithmetic.
c) Students who have been taught to rely on intuitive arguments like this often make mistakes (e.g. the student who tries to use chocolate bar explanations to convince themselves that 0/0=0; I’ve literally seen this sort of nonsense from my students in 1st year calculus)
d) The idea that students who have spent loads of time on these intuitive picture explanations will be better at fraction arithmetic is absurd. Fraction arithmetic needs to be practiced so much it’s automatic. Any student who resorts to pictures to figure out why you need to flip and multiply in the middle of an algebra problem is not fluent with fraction arithmetic and is bound to struggle. So, this begs the question, why spend so much time on chocolate bar fraction explanations?
e) Regarding point d), I’ve seen this on university entrance exams. Some students have spent more time on picture explanations than actually practicing math and will draw tick marks and circle groupings to figure out 24/8. Believe me, they’re spending time on this in school. I see the results.
f) Many elementary teachers would find this lesson challenging to give. I suspect both the teacher and students would be confused. It’s also time consuming to prepare a lesson like this and think of all the possibilities, etc. Then the teachers get blamed when it doesn’t work out. No, it doesn’t need to be this complicated in the first place. I think scripted lessons are the best way to go in elementary school.
g) I suppose we’re supposed to believe this is “understanding.” However, this is simply intuitive reasoning using pictures that works only in certain situations. If we are really serious about understanding then we’d have to discuss the algebraic reasoning, but for some reason that never makes it into the conversation.
Brooke’s data and my own internal data using a similar approach both speak to the success of a teaching method like this. You’re exactly right that it’s very hard to do, which is the challenge I name in this article and grapple with regularly.
Similarly to the fact fluency challenges, when students are taught inefficient strategies in one grade to build conceptual understanding of the concept they must move to more efficient strategies in future grades. It sounds like what you’re seeing is that many students are still using extremely inefficient strategies and never made the switch to more efficient strategies. In my experience, this comes from students not having enough practice time with the more efficient strategies. My experience has shown that when students see the connection between a less efficient strategy or more concrete representation and then get sufficient time to practice the more efficient/more abstract approach, their overall understanding is stronger. Brooke does an excellent job of connecting strategies and providing sufficient practice time (which wasn’t shown in the video referenced).
Does it show that this particular method - effectively productive struggle, which hasn't been shown to be effective in any large-scale trials - works? Was there a control group? How was the study designed? Is this published in a peer-reviewed journal somewhere?
There is a research base in the science of learning for effortful thinking tasks. From Deans for Impact, “Teachers can prompt deeper, ‘effortful’ thinking with elaborative questions and tasks that cue thinking about relationships between ideas. These prompts often start with ‘how’ or ‘why’.” (https://www.deansforimpact.org/files/assets/lbsdanchorcharteft.pdf )
This is what is happening in the video: the teacher has given the students a task for which they have all the necessary prior knowledge to solve (understanding of the concept of fractions of a set and procedural fluency with fractions/decimals). The task is bringing together two pieces of knowledge students already have and cueing them to think about the relationship between these ideas. A later progression in this understanding is connecting it to the algebraic approach for solving.
There is also a strong research base for spaced, interleaved practice which is also a key component of this teaching model (not shown in the specific portion of the lesson videoed). I’m not aware of any research into the specific combination of an effortful thinking task and spaced, interleaved practice but the research base for both individually is, to my knowledge, robust.
It is my opinion that the combination of these research-based, science of learning practices is what has led to strong student achievement data at Brooke (in the video) and at the schools where I replicated this model. Though it is essential to underscore, as you also point out, this is very hard to do and requires an immense amount of teacher training.
"There is also a strong research base for spaced, interleaved practice which is also a key component of this teaching model" True. If the school is using spaced, interleaved and repeated practice in an effective way that's great and can have a huge impact. The video, though - not convinced that's what is meant by desirable difficulties (especially given this went on for a week and this particular exercise offers little beyond the previous ones mentioned, in my opinion). Thanks for the discussion :)
I’m not a math teacher, but I found the video lesson very interesting. A lot seemed useful to me but also somewhat inefficient. I wonder how this concept is taught in Direct Instruction programs.
That's a good question, Miriam. The reason I wanted to show the Brooke video and talk about it (and if you're interested, watch some of the others, too, like the kinder/first grade videos are fascinating: https://resources.ebrooke.org/#/elements/math/), is because their math outcomes are very good--so they've figured something out. It's my understanding that they're very intentional about connecting each new concept to the one they previously learned, so their knowledge is meaningful, which might look inefficient at first. But, according to Boyd, they also do a good job solidifying that learning and then turning to practice to get it locked in.
Miriam expressed my thoughts exactly. Watching a sixth grade Brooke lesson and the lesson you linked, I watched the kids flounder around, trying to notice what the teacher intended for them to notice, and only really getting there after the teacher asked more directed questions. I can see how guided inquiry can be useful, but I think it needs to be even more narrowly structured than the lessons shown. Why not start with the more directed questions?
The teacher says in the video she wanted to know what students would do if she actually gave them 9 chocolate bars. She's moving them from picturing how to divide the set visually first, using concrete objects, before moving to the algorithm. This makes a lot of sense to me--she wants to make sure students understand exactly what they're dividing before they do it with numbers, right? And she wants them, if possible, to make that leap themselves. I'm not a teacher, so I don't know whether it should be more narrowly structured. Would love to hear from others!
I want to address a few things regarding the interview in your post "How to Teach Math". I'll address the video in another post, but I'd like to start with the basic fact recommendation.
I'd like to address this, which I'm quoting from your "How to Teach Math" interview:
"But what research shows (take a look here and here) is that middle phase—I've started to build some automaticity with really easy facts, like the plus 1 facts or doubles facts, and I can use those to get to the answer for the facts I have not memorized. It's faster than counting. The middle phase, then, is how you get to the automaticity, in a stronger way."
Actually, I interviewed Brian Poncy - a basic fact researcher - and he cited the Vasilyeva paper, which is one of the articles cited in the above quote. He notes that the focus on strategies to get to the automaticity stage is likely not that helpful. In fact, that paper may provide some evidence for the opposite: that if you have your math facts memorized, it's going to be easier to decompose larger numbers. I would suggest anyone reading this look into Brian Poncy's program Facts on Fire (or M.I.N.D.) for helping kids become fluent with math facts. I'm going to paste part of that interview below, but the full transcript is here: annastokke.com/ep-29-transcript. This is a (long) quote from Brian Poncy in that interview.
"NCTM is all about memorizing stuff. Just some stuff and in some ways. For example, if you're going to use a decomposition strategy, you have to have your doubles memorized... And so they're cool with memorizing that, and they're cool memorizing the strategy of doubles plus or minus one, and they're cool with the procedure of, “Okay, do I want to go up one or down one?” And so, people that say, “If they memorize stuff, they're just not going to get the concept.”
It's like, we'll quit talking out of both sides of your mouth. And then when we get to tens, now they memorize all the tens and now we can start making tens because I know seven plus three and four plus six and all that stuff. And now you get this little set of facts on each side that doubles plus and minus one and then making pens don't really apply to And then kids usually will revert back to good old count-up strategies....
In Taiwan, 63 percent of the kids used a decomposition strategy as compared to only 22 that used accounting strategy and when they went to double digits 75 percent of Taiwanese kids used decomposition, whereas only 50 percent of American kids. Only 10 percent of Taiwanese kids used counting procedures and almost 40 percent of American kids. And so if you I know it's a lot of data but you know when I sit back and I thought about it retrieval with basic facts supports decomposition strategy use.
And as more kids memorize those single-digit skills or those problems, it allows them to actually achieve the very goals that the reformists are wanting to do. And so it's interesting, when we talked earlier, I was like, [quoting reformers] “Well, as long as they memorize things through procedures, they're okay.”
But what we're doing is we're making kids count and use count on procedures and the temporal window does not allow them to basically make that a memorized fact. Therefore, that never moves to a decomposition issue. And so what Taiwan does, and Asian countries are more apt to do, is have them learn and memorize those core facts and they use the declarative fact to basically inform the procedure.
In America, NCTM wants to focus on the procedure to build the fact. And so, your lower kids get left in the dust. And so, you know, you got this great organization that's championing diversity and all this stuff. But in reality, the most vulnerable populations are the very ones that are getting hurt by the pedagogies they promote."
Thanks for taking the time to read and respond, Anna. The study you reference shows that Taiwanese students use a higher level of both decomposition and retrieval with single-digit facts and that this benefits them in more easily using decomposition with larger addition problems. It also names that US students were primarily using counting strategies and didn’t progress to more sophisticated decomposition and retrieval strategies, which then impacts their ability to add the larger numbers. The study names that Taiwanese curricula are more targeted in their teaching of decomposition with single-digit facts than US curricula.
Arthur Baroody’s research, which is also cited in this study, shows that these strategies are a progression. Students start with the least efficient strategies, counting strategies (count all and count on), then move to decomposition strategies (using related facts, like I describe in the article). From there, students need to progress from decomposition strategies to retrieval. You and I are in agreement that too many US students never get to retrieval. My opinion based on this research is that it is because US schools don’t systematically emphasize decomposition strategies and then, from there, provide enough practice to reach the retrieval stage. (The authors of the study you reference and linked in this interview also have another study titled, “Arithmetic Accuracy in Children From High- and Low-Income Schools: What Do Strategies Have to Do With It?” in which they show that students using more sophisticated decomposition and retrieval strategies have higher accuracy with addition.)
I don't think that's a correct interpretation of the results in the Vasilyeva paper. Regarding decomposition strategies, the study aimed 1) to explore strategies used to solve complex arithmetic problems and compare them with the strategies used on simple problems and 2) to compare the use of strategies by American and Taiwanese students. What they found was that more Taiwanese students were able to recall single-digit basic facts from memory (and in general more quickly) than American students, who tended to rely on counting strategies, and that more Taiwanese students were able to use decomposition strategies with double-digit numbers, resulting in more accuracy with double-digit arithmetic (which is really when we'd ultimately want students to use decomposition strategies - not with single digits). That makes sense because knowing single-digit facts cold frees up working memory so that one can use a decomposition strategy for larger numbers.
What Poncy was saying is that focussing on a procedure to inform a single-digit fact, often just ends up in kids using count-on strategies (even if said goal is doubles strategies, etc.) but when committing things to memory there has to be a tight temporal window between the stimulus and response. If someone is using a strategy to get the fact, the temporal window becomes too great and they don't pair the stimulus with the response. E.g. 9+6 has to be paired with 15. That's why things like cover-copy-compare and flash cards are actually really helpful for memorizing single-digit math facts. So, if we aim to get students to fluency with basic facts (which means both accuracy and speed), decomposition strategies are likely not the most effective way to do that.
Yes, the study shows that the Taiwanese students were using retrieval of single-digit facts to more easily decompose two-digit addition. By the time of the study, the majority of Taiwanese students were already at the retrieval stage with single-digit facts so the study itself isn’t about how single-digit retrieval strategies were built. However, there are several indicators in the study as to how Taiwanese students built fluency with single-digit retrieval strategies:
(1) The intro to the study highlights differences in how US and Asian students solve single-digit addition problems: “Geary and colleagues (1996) demonstrated cross-national differences in strategies used for single-digit addition. At the start of first grade, Chinese students relied mostly on decomposition and retrieval, whereas American students relied mostly on counting. At the end of first grade, the prevalent strategy among Chinese students became retrieval, whereas a large percentage of Americans continued to use counting.”
(2) This difference in strategy use was also seen in curricular resources: “Another notable aspect of early math instruction for which cross-national differences have been documented is the emphasis on decomposition strategy. Although first graders in the United States, as well as in Asian countries, are introduced to a variety of arithmetic strategies, including retrieval, counting and decomposition, there is a clear difference in the extent to which particular strategies are emphasized. Our examination of curricular materials in the countries participating in the present study revealed that in Taiwan instruction emphasized the use of decomposition, whereas in the United States there was no particular emphasis on one strategy over the other. Children in both countries are taught decomposition through doubles (e.g.,6+7=[6+6]+1=12+1) and through 10 (e.g., 6+7=[6+4]+3=10+3), but there is more instruction on base-10 decomposition in Taiwan compared with that in the United States.”
(3) The definition given of single-digit computational fluency is one that cites the Baroody research for how meaningful memorization is built through the strategy progression: “Computational fluency is attained through accumulated practice solving simple problems, which leads to memorizing basic number facts and the development of interconnected numeric knowledge (Baroody, 2006).” The Baroody research is clear that the decomposition strategies do matter in building this “interconnected numeric knowledge.”
This, combined with the other research I cited in my earlier comment, underscore the importance of building a connected understanding of single-digit facts through first counting strategies, then decomposition, and then retrieval. We are in complete agreement, however, that the timing of this process is very important. Based on the earlier quotes from the study, there appears to be a systematic focus in many Asian countries on moving students along this continuum in a way that US schools aren’t currently doing.
But the cited papers don't support the claim b/c they don't directly study how to best get kids to memorize multiplication facts. Note that the Baroody paper isn't a research publication - it's an NCTM publication.
Again, thanks for the discussion. We do agree that kids need to memorize math facts, and earlier than is currently the norm. I'd like to leave a link to Brian Poncy's program, for anyone reading, since he does research in this area. The M.I.N.D is an empirically-validated program and it's free. https://brianponcy.wixsite.com/mind/intervention-resources
"Guided inquiry" is one of those terms that is easily misinterpreted so that some Ts will practice minimal guidance and think they are practicing what is described in the interview. Let's call it explicit instruction, with the understanding that that encompasses some leaps in reasoning based on the incremental steps involved with proper scaffolding of problems. That's what JUMP Math does. Calling it any kind of "inquiry" opens the door to ineffective teaching practices.
At the same time, I get what Boyd is saying here, that you also don't want teachers over-correcting and only teaching procedures—which is what led the conceptual revolution in the first place. I'm less hung up on what we call it and more hung up on how more teachers can teach effectively!
The video. I don’t know what the situation is in this particular school - you say they get good results - but I would not recommend doing this at all.
This class previously spent a week doing similar activities to work out things like ¾ x 6=8. Now they are spending a long painful class trying to figure out ¾ x 9 in the same way. Why? What next? Another week working out 2/17 x 5? Yes, I know this is supposed to be “understanding.” No, it’s just a complicated way of working out a problem using pictures.
You can demonstrate some simple examples for students using pictures to get the point across, but this doesn’t need to go on for so long. We should be trying to make things less complicated for students, not more complicated.
Some points:
a) This approach is confusing and not all students will be able to follow this. Even in this class, not all students are participating. Note the one boy (~2:30) who answers something incorrectly (the teacher doesn’t correct him?) and then puts his head down. I highly doubt this is going to work out well in your average class of 5th graders. Some students get to be the stars that come up with all the ideas; other students just feel stupid.
b) This is incredibly time consuming. If there were huge benefits to this, that may be reasonable. I’m not convinced. Students would be better off spending more time practicing fraction arithmetic.
c) Students who have been taught to rely on intuitive arguments like this often make mistakes (e.g. the student who tries to use chocolate bar explanations to convince themselves that 0/0=0; I’ve literally seen this sort of nonsense from my students in 1st year calculus)
d) The idea that students who have spent loads of time on these intuitive picture explanations will be better at fraction arithmetic is absurd. Fraction arithmetic needs to be practiced so much it’s automatic. Any student who resorts to pictures to figure out why you need to flip and multiply in the middle of an algebra problem is not fluent with fraction arithmetic and is bound to struggle. So, this begs the question, why spend so much time on chocolate bar fraction explanations?
e) Regarding point d), I’ve seen this on university entrance exams. Some students have spent more time on picture explanations than actually practicing math and will draw tick marks and circle groupings to figure out 24/8. Believe me, they’re spending time on this in school. I see the results.
f) Many elementary teachers would find this lesson challenging to give. I suspect both the teacher and students would be confused. It’s also time consuming to prepare a lesson like this and think of all the possibilities, etc. Then the teachers get blamed when it doesn’t work out. No, it doesn’t need to be this complicated in the first place. I think scripted lessons are the best way to go in elementary school.
g) I suppose we’re supposed to believe this is “understanding.” However, this is simply intuitive reasoning using pictures that works only in certain situations. If we are really serious about understanding then we’d have to discuss the algebraic reasoning, but for some reason that never makes it into the conversation.
Brooke’s data and my own internal data using a similar approach both speak to the success of a teaching method like this. You’re exactly right that it’s very hard to do, which is the challenge I name in this article and grapple with regularly.
Similarly to the fact fluency challenges, when students are taught inefficient strategies in one grade to build conceptual understanding of the concept they must move to more efficient strategies in future grades. It sounds like what you’re seeing is that many students are still using extremely inefficient strategies and never made the switch to more efficient strategies. In my experience, this comes from students not having enough practice time with the more efficient strategies. My experience has shown that when students see the connection between a less efficient strategy or more concrete representation and then get sufficient time to practice the more efficient/more abstract approach, their overall understanding is stronger. Brooke does an excellent job of connecting strategies and providing sufficient practice time (which wasn’t shown in the video referenced).
Does it show that this particular method - effectively productive struggle, which hasn't been shown to be effective in any large-scale trials - works? Was there a control group? How was the study designed? Is this published in a peer-reviewed journal somewhere?
There is a research base in the science of learning for effortful thinking tasks. From Deans for Impact, “Teachers can prompt deeper, ‘effortful’ thinking with elaborative questions and tasks that cue thinking about relationships between ideas. These prompts often start with ‘how’ or ‘why’.” (https://www.deansforimpact.org/files/assets/lbsdanchorcharteft.pdf )
This is what is happening in the video: the teacher has given the students a task for which they have all the necessary prior knowledge to solve (understanding of the concept of fractions of a set and procedural fluency with fractions/decimals). The task is bringing together two pieces of knowledge students already have and cueing them to think about the relationship between these ideas. A later progression in this understanding is connecting it to the algebraic approach for solving.
There is also a strong research base for spaced, interleaved practice which is also a key component of this teaching model (not shown in the specific portion of the lesson videoed). I’m not aware of any research into the specific combination of an effortful thinking task and spaced, interleaved practice but the research base for both individually is, to my knowledge, robust.
It is my opinion that the combination of these research-based, science of learning practices is what has led to strong student achievement data at Brooke (in the video) and at the schools where I replicated this model. Though it is essential to underscore, as you also point out, this is very hard to do and requires an immense amount of teacher training.
"There is also a strong research base for spaced, interleaved practice which is also a key component of this teaching model" True. If the school is using spaced, interleaved and repeated practice in an effective way that's great and can have a huge impact. The video, though - not convinced that's what is meant by desirable difficulties (especially given this went on for a week and this particular exercise offers little beyond the previous ones mentioned, in my opinion). Thanks for the discussion :)
I’m not a math teacher, but I found the video lesson very interesting. A lot seemed useful to me but also somewhat inefficient. I wonder how this concept is taught in Direct Instruction programs.
That's a good question, Miriam. The reason I wanted to show the Brooke video and talk about it (and if you're interested, watch some of the others, too, like the kinder/first grade videos are fascinating: https://resources.ebrooke.org/#/elements/math/), is because their math outcomes are very good--so they've figured something out. It's my understanding that they're very intentional about connecting each new concept to the one they previously learned, so their knowledge is meaningful, which might look inefficient at first. But, according to Boyd, they also do a good job solidifying that learning and then turning to practice to get it locked in.
Miriam expressed my thoughts exactly. Watching a sixth grade Brooke lesson and the lesson you linked, I watched the kids flounder around, trying to notice what the teacher intended for them to notice, and only really getting there after the teacher asked more directed questions. I can see how guided inquiry can be useful, but I think it needs to be even more narrowly structured than the lessons shown. Why not start with the more directed questions?
The teacher says in the video she wanted to know what students would do if she actually gave them 9 chocolate bars. She's moving them from picturing how to divide the set visually first, using concrete objects, before moving to the algorithm. This makes a lot of sense to me--she wants to make sure students understand exactly what they're dividing before they do it with numbers, right? And she wants them, if possible, to make that leap themselves. I'm not a teacher, so I don't know whether it should be more narrowly structured. Would love to hear from others!
I want to address a few things regarding the interview in your post "How to Teach Math". I'll address the video in another post, but I'd like to start with the basic fact recommendation.
I'd like to address this, which I'm quoting from your "How to Teach Math" interview:
"But what research shows (take a look here and here) is that middle phase—I've started to build some automaticity with really easy facts, like the plus 1 facts or doubles facts, and I can use those to get to the answer for the facts I have not memorized. It's faster than counting. The middle phase, then, is how you get to the automaticity, in a stronger way."
Actually, I interviewed Brian Poncy - a basic fact researcher - and he cited the Vasilyeva paper, which is one of the articles cited in the above quote. He notes that the focus on strategies to get to the automaticity stage is likely not that helpful. In fact, that paper may provide some evidence for the opposite: that if you have your math facts memorized, it's going to be easier to decompose larger numbers. I would suggest anyone reading this look into Brian Poncy's program Facts on Fire (or M.I.N.D.) for helping kids become fluent with math facts. I'm going to paste part of that interview below, but the full transcript is here: annastokke.com/ep-29-transcript. This is a (long) quote from Brian Poncy in that interview.
"NCTM is all about memorizing stuff. Just some stuff and in some ways. For example, if you're going to use a decomposition strategy, you have to have your doubles memorized... And so they're cool with memorizing that, and they're cool memorizing the strategy of doubles plus or minus one, and they're cool with the procedure of, “Okay, do I want to go up one or down one?” And so, people that say, “If they memorize stuff, they're just not going to get the concept.”
It's like, we'll quit talking out of both sides of your mouth. And then when we get to tens, now they memorize all the tens and now we can start making tens because I know seven plus three and four plus six and all that stuff. And now you get this little set of facts on each side that doubles plus and minus one and then making pens don't really apply to And then kids usually will revert back to good old count-up strategies....
In Taiwan, 63 percent of the kids used a decomposition strategy as compared to only 22 that used accounting strategy and when they went to double digits 75 percent of Taiwanese kids used decomposition, whereas only 50 percent of American kids. Only 10 percent of Taiwanese kids used counting procedures and almost 40 percent of American kids. And so if you I know it's a lot of data but you know when I sit back and I thought about it retrieval with basic facts supports decomposition strategy use.
And as more kids memorize those single-digit skills or those problems, it allows them to actually achieve the very goals that the reformists are wanting to do. And so it's interesting, when we talked earlier, I was like, [quoting reformers] “Well, as long as they memorize things through procedures, they're okay.”
But what we're doing is we're making kids count and use count on procedures and the temporal window does not allow them to basically make that a memorized fact. Therefore, that never moves to a decomposition issue. And so what Taiwan does, and Asian countries are more apt to do, is have them learn and memorize those core facts and they use the declarative fact to basically inform the procedure.
In America, NCTM wants to focus on the procedure to build the fact. And so, your lower kids get left in the dust. And so, you know, you got this great organization that's championing diversity and all this stuff. But in reality, the most vulnerable populations are the very ones that are getting hurt by the pedagogies they promote."
Thanks for taking the time to read and respond, Anna. The study you reference shows that Taiwanese students use a higher level of both decomposition and retrieval with single-digit facts and that this benefits them in more easily using decomposition with larger addition problems. It also names that US students were primarily using counting strategies and didn’t progress to more sophisticated decomposition and retrieval strategies, which then impacts their ability to add the larger numbers. The study names that Taiwanese curricula are more targeted in their teaching of decomposition with single-digit facts than US curricula.
Arthur Baroody’s research, which is also cited in this study, shows that these strategies are a progression. Students start with the least efficient strategies, counting strategies (count all and count on), then move to decomposition strategies (using related facts, like I describe in the article). From there, students need to progress from decomposition strategies to retrieval. You and I are in agreement that too many US students never get to retrieval. My opinion based on this research is that it is because US schools don’t systematically emphasize decomposition strategies and then, from there, provide enough practice to reach the retrieval stage. (The authors of the study you reference and linked in this interview also have another study titled, “Arithmetic Accuracy in Children From High- and Low-Income Schools: What Do Strategies Have to Do With It?” in which they show that students using more sophisticated decomposition and retrieval strategies have higher accuracy with addition.)
I don't think that's a correct interpretation of the results in the Vasilyeva paper. Regarding decomposition strategies, the study aimed 1) to explore strategies used to solve complex arithmetic problems and compare them with the strategies used on simple problems and 2) to compare the use of strategies by American and Taiwanese students. What they found was that more Taiwanese students were able to recall single-digit basic facts from memory (and in general more quickly) than American students, who tended to rely on counting strategies, and that more Taiwanese students were able to use decomposition strategies with double-digit numbers, resulting in more accuracy with double-digit arithmetic (which is really when we'd ultimately want students to use decomposition strategies - not with single digits). That makes sense because knowing single-digit facts cold frees up working memory so that one can use a decomposition strategy for larger numbers.
What Poncy was saying is that focussing on a procedure to inform a single-digit fact, often just ends up in kids using count-on strategies (even if said goal is doubles strategies, etc.) but when committing things to memory there has to be a tight temporal window between the stimulus and response. If someone is using a strategy to get the fact, the temporal window becomes too great and they don't pair the stimulus with the response. E.g. 9+6 has to be paired with 15. That's why things like cover-copy-compare and flash cards are actually really helpful for memorizing single-digit math facts. So, if we aim to get students to fluency with basic facts (which means both accuracy and speed), decomposition strategies are likely not the most effective way to do that.
Yes, the study shows that the Taiwanese students were using retrieval of single-digit facts to more easily decompose two-digit addition. By the time of the study, the majority of Taiwanese students were already at the retrieval stage with single-digit facts so the study itself isn’t about how single-digit retrieval strategies were built. However, there are several indicators in the study as to how Taiwanese students built fluency with single-digit retrieval strategies:
(1) The intro to the study highlights differences in how US and Asian students solve single-digit addition problems: “Geary and colleagues (1996) demonstrated cross-national differences in strategies used for single-digit addition. At the start of first grade, Chinese students relied mostly on decomposition and retrieval, whereas American students relied mostly on counting. At the end of first grade, the prevalent strategy among Chinese students became retrieval, whereas a large percentage of Americans continued to use counting.”
(2) This difference in strategy use was also seen in curricular resources: “Another notable aspect of early math instruction for which cross-national differences have been documented is the emphasis on decomposition strategy. Although first graders in the United States, as well as in Asian countries, are introduced to a variety of arithmetic strategies, including retrieval, counting and decomposition, there is a clear difference in the extent to which particular strategies are emphasized. Our examination of curricular materials in the countries participating in the present study revealed that in Taiwan instruction emphasized the use of decomposition, whereas in the United States there was no particular emphasis on one strategy over the other. Children in both countries are taught decomposition through doubles (e.g.,6+7=[6+6]+1=12+1) and through 10 (e.g., 6+7=[6+4]+3=10+3), but there is more instruction on base-10 decomposition in Taiwan compared with that in the United States.”
(3) The definition given of single-digit computational fluency is one that cites the Baroody research for how meaningful memorization is built through the strategy progression: “Computational fluency is attained through accumulated practice solving simple problems, which leads to memorizing basic number facts and the development of interconnected numeric knowledge (Baroody, 2006).” The Baroody research is clear that the decomposition strategies do matter in building this “interconnected numeric knowledge.”
This, combined with the other research I cited in my earlier comment, underscore the importance of building a connected understanding of single-digit facts through first counting strategies, then decomposition, and then retrieval. We are in complete agreement, however, that the timing of this process is very important. Based on the earlier quotes from the study, there appears to be a systematic focus in many Asian countries on moving students along this continuum in a way that US schools aren’t currently doing.
But the cited papers don't support the claim b/c they don't directly study how to best get kids to memorize multiplication facts. Note that the Baroody paper isn't a research publication - it's an NCTM publication.
Again, thanks for the discussion. We do agree that kids need to memorize math facts, and earlier than is currently the norm. I'd like to leave a link to Brian Poncy's program, for anyone reading, since he does research in this area. The M.I.N.D is an empirically-validated program and it's free. https://brianponcy.wixsite.com/mind/intervention-resources
"Guided inquiry" is one of those terms that is easily misinterpreted so that some Ts will practice minimal guidance and think they are practicing what is described in the interview. Let's call it explicit instruction, with the understanding that that encompasses some leaps in reasoning based on the incremental steps involved with proper scaffolding of problems. That's what JUMP Math does. Calling it any kind of "inquiry" opens the door to ineffective teaching practices.
At the same time, I get what Boyd is saying here, that you also don't want teachers over-correcting and only teaching procedures—which is what led the conceptual revolution in the first place. I'm less hung up on what we call it and more hung up on how more teachers can teach effectively!