In Virginia - and the rest of the country is little different - the standardized math test starts in the 3rd grade and is given every year. In the 3rd grade the test is 40% without calculator and 60% with calculator. Each year the non-calculator portion decreases until in the 8th grade (and ALL of high school) the test is 100% calculator active. More significant, there are NO NUMERICAL CALCULATIONS ON ANY NON-CALCULATOR PORTION OF ANY TEST! Because the pressure on teachers to have a high pass rate on the standardized test is IMMENSE, they teach ALL elementary students to do all numerical work on the calculator. The result is we have been producing innumerate students for almost twenty years. I taught high school math for 36 years and the deterioration of number sense in the students I taught was dramatic. An example: In a class of 30 Algebra I students the topic was factoring an expression. This required them to choose the correct pair of factors of 63. When asked the entire class could come up with only one pair of numbers that multiplied to give 63; 1 and 63. Clearly nobody knew their multiplication tables for 7 or 9; not to mention the idea that if 3 divides both 6 and 3, it must divide 63. If you look up Cognitive Load Theory, you will find research that says our short-term working memory is very limited - we can hold four facts for about a minute - and when it takes more than a minute to determine this one piece of information needed for a multi-step process like factoring, the high probability is the student will experience cognitive overload, for which the most common result is the student quits trying - a result ALL math teachers are quite familiar with! Since there is no high school diploma available for a student who can't pass Algebra I, the most likely result was a significant portion of my Algebra I class would never graduate.
The answer is disturbingly simple - remove the calculators from the elementary classrooms during instruction of numerical calculating skills.
Math for elementary students is indeed for solving problems. Those problems can include questions about quantities, relative distances or rates, areas, and volumes of relatively simple solids/containers.
There are LOTS of opportunities to solve this type of problems within a framework of everyday cooking, travel, shopping, and house/yard work. Teachers could learn to provide example problems using situations common to their area (eg. raking leaves in suburbs or rural areas, walking 5 blocks (where a block is X feet or yards, or meters long) in big cities, planning a party menu for X people with $Y to spend, etc.)
Not to mention explicitly teaching kids to translate rates into distance/time for all kinds of time increments, and how to calculate a rate in different distance or time units from those given.
Yes, there should be explicit instruction with worked examples of particular problem types, and then give students scaffolded problems that increment in difficulty. Problems in text books often become too complex too fast, and it is hard for students to connect the initial methods to more complex versions.
Singapore's "Primary Math" series does an excellent job of providing such a structure, and provide multi-step problems so that students employ a number of procedures. This builds mathematical reasoning on how to solve a variety of problems. An example of a multi-step problem is: "John buys 2 pencils that sell for 3 for 15 cents and 4 notebooks that sell for 5 for $2.00. He pays with a five dollar bill. How much change does he receive?"
On the issue of productive struggle, Amanda Vanderheyden addresses this. Having students "struggle" with a problem when they are in the acquisition stage of learning is too early, and will be "unproductive". She has said that struggle is "productive" when information is in the "Generalization and adaptation" stage, rather than the "acquisition" stage. She talks about this at 23:00 to 25:00 minutes in the interview with Anna Stokke on her Chalk and Talk podcast. https://chalkandtalkpodcast.podbean.com/e/ep-3-the-science-of-math-with-amanda-vanderheyden/
In short, it isn't so much that there's too much "explicit instruction" with no opportunity for applying the procedures, it's as Amanda says: "Generalization instruction has to be for a skill for which children have reached mastery. ...It's not that teachers are using ineffective tactics, period. It's, they're using them at the wrong time."
You’re so right that the lack of intentionally sequenced problems in many text books/curricula is a massive issue. It’s so disheartening to watch students get frustrated trying to work through problems that don’t gradually build in rigor (and such a waste of teacher time to tell students, “Solve the problems in this order: 4, 1, 2, 7, 9… when it should have already been written in that order anyway.)
I think a lot about the analogy to sports and music and how we don’t hold children back from playing the game/playing a song until they’ve reached full mastery of the basic skills. There’s a fluid motion between practicing skills and trying to put them together (but putting them together inefficiently which necessitates more practice with skills). I think this gets back to your earlier point about the need for scaffolding in practice problems. We don’t let 5-year-olds who are just learning to play basketball go play a game against a middle school team. Everyone would agree that would be a massive disaster and fully destroy and burgeoning love of basketball for those 5-year-olds. But we do let them play a game against other 5-year-olds, knowing it will be inefficient. Intentionality in the way we scaffold “playing the game of math” is essential for ensuring that the practice-play cycle continues along productively.
I vaguely remember elementary school math. We learned to figure out how many apples Dick and Jane had, If Dick had five and Jane had seven. And how many would each have, if they shared them equally? To a six year-old, learning how to make such determinations is significant. If math is made relevant to the student, they will absorb it.
Math is not just numbers; it is fundamental rational thought. Every child should learn the thought process, beyond just crunching numbers.
In Virginia - and the rest of the country is little different - the standardized math test starts in the 3rd grade and is given every year. In the 3rd grade the test is 40% without calculator and 60% with calculator. Each year the non-calculator portion decreases until in the 8th grade (and ALL of high school) the test is 100% calculator active. More significant, there are NO NUMERICAL CALCULATIONS ON ANY NON-CALCULATOR PORTION OF ANY TEST! Because the pressure on teachers to have a high pass rate on the standardized test is IMMENSE, they teach ALL elementary students to do all numerical work on the calculator. The result is we have been producing innumerate students for almost twenty years. I taught high school math for 36 years and the deterioration of number sense in the students I taught was dramatic. An example: In a class of 30 Algebra I students the topic was factoring an expression. This required them to choose the correct pair of factors of 63. When asked the entire class could come up with only one pair of numbers that multiplied to give 63; 1 and 63. Clearly nobody knew their multiplication tables for 7 or 9; not to mention the idea that if 3 divides both 6 and 3, it must divide 63. If you look up Cognitive Load Theory, you will find research that says our short-term working memory is very limited - we can hold four facts for about a minute - and when it takes more than a minute to determine this one piece of information needed for a multi-step process like factoring, the high probability is the student will experience cognitive overload, for which the most common result is the student quits trying - a result ALL math teachers are quite familiar with! Since there is no high school diploma available for a student who can't pass Algebra I, the most likely result was a significant portion of my Algebra I class would never graduate.
The answer is disturbingly simple - remove the calculators from the elementary classrooms during instruction of numerical calculating skills.
Math for elementary students is indeed for solving problems. Those problems can include questions about quantities, relative distances or rates, areas, and volumes of relatively simple solids/containers.
There are LOTS of opportunities to solve this type of problems within a framework of everyday cooking, travel, shopping, and house/yard work. Teachers could learn to provide example problems using situations common to their area (eg. raking leaves in suburbs or rural areas, walking 5 blocks (where a block is X feet or yards, or meters long) in big cities, planning a party menu for X people with $Y to spend, etc.)
Not to mention explicitly teaching kids to translate rates into distance/time for all kinds of time increments, and how to calculate a rate in different distance or time units from those given.
Yes, there should be explicit instruction with worked examples of particular problem types, and then give students scaffolded problems that increment in difficulty. Problems in text books often become too complex too fast, and it is hard for students to connect the initial methods to more complex versions.
Singapore's "Primary Math" series does an excellent job of providing such a structure, and provide multi-step problems so that students employ a number of procedures. This builds mathematical reasoning on how to solve a variety of problems. An example of a multi-step problem is: "John buys 2 pencils that sell for 3 for 15 cents and 4 notebooks that sell for 5 for $2.00. He pays with a five dollar bill. How much change does he receive?"
On the issue of productive struggle, Amanda Vanderheyden addresses this. Having students "struggle" with a problem when they are in the acquisition stage of learning is too early, and will be "unproductive". She has said that struggle is "productive" when information is in the "Generalization and adaptation" stage, rather than the "acquisition" stage. She talks about this at 23:00 to 25:00 minutes in the interview with Anna Stokke on her Chalk and Talk podcast. https://chalkandtalkpodcast.podbean.com/e/ep-3-the-science-of-math-with-amanda-vanderheyden/
In short, it isn't so much that there's too much "explicit instruction" with no opportunity for applying the procedures, it's as Amanda says: "Generalization instruction has to be for a skill for which children have reached mastery. ...It's not that teachers are using ineffective tactics, period. It's, they're using them at the wrong time."
You’re so right that the lack of intentionally sequenced problems in many text books/curricula is a massive issue. It’s so disheartening to watch students get frustrated trying to work through problems that don’t gradually build in rigor (and such a waste of teacher time to tell students, “Solve the problems in this order: 4, 1, 2, 7, 9… when it should have already been written in that order anyway.)
I think a lot about the analogy to sports and music and how we don’t hold children back from playing the game/playing a song until they’ve reached full mastery of the basic skills. There’s a fluid motion between practicing skills and trying to put them together (but putting them together inefficiently which necessitates more practice with skills). I think this gets back to your earlier point about the need for scaffolding in practice problems. We don’t let 5-year-olds who are just learning to play basketball go play a game against a middle school team. Everyone would agree that would be a massive disaster and fully destroy and burgeoning love of basketball for those 5-year-olds. But we do let them play a game against other 5-year-olds, knowing it will be inefficient. Intentionality in the way we scaffold “playing the game of math” is essential for ensuring that the practice-play cycle continues along productively.
I’m of the opinion we should use explicit instruction pretty much all the time. Included in that is giving challenging problems and fading guidance.
I vaguely remember elementary school math. We learned to figure out how many apples Dick and Jane had, If Dick had five and Jane had seven. And how many would each have, if they shared them equally? To a six year-old, learning how to make such determinations is significant. If math is made relevant to the student, they will absorb it.
Math is not just numbers; it is fundamental rational thought. Every child should learn the thought process, beyond just crunching numbers.
Forgot to mention: Christian Bohkove is not a cognitive scientist. He is a math education professor: https://bokhove.net/about/
Yep good catch, Barry, correcting that now!