Boaler lists several misconceptions on page 31(of Mathematical Mind...

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Boaler lists several misconceptions on page 31(of Mathematical Mindsets Summary, Chapter 3: The Creativity and Beauty in Mathematics) “that math is a subject of rules and procedures, that being good at math means being fast at math, that math is all about certainty and right and wrong answers, and that math is all about numbers.” Choose two of the misconceptions on this list and write a response for each. Why is it a misconception, what truth could it be replaced by, and how would you support your assertions?
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Misconception: “being good at math means being fast at math”
It is a misconception because it is wrong.
Truth: Being fast is not good enough and in fact is not really needed.
Support: Fast calculations these days are fully automated - this is what computers are for. The great thinkers make connections, think logically, and use clear sense and data creatively
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      • Illuminate\Cache\Events\CacheHit (219μs)
      • Illuminate\Cache\Events\CacheHit (161μs)
      • Illuminate\Cache\Events\CacheHit (179μs)
      • Illuminate\Cache\Events\CacheHit (158μs)
      • Illuminate\Cache\Events\CacheHit (181μs)
      • Illuminate\Cache\Events\CacheHit (153μs)
      • creating: site.layouts.app (483μs)
      • composing: site.layouts.app (21μs)
      • creating: components.canonical (418μs)
      • composing: components.canonical (102μs)
      • creating: components.open-graph (215μs)
      • composing: components.open-graph (78μs)
      • creating: site.headers.header (321μs)
      • composing: site.headers.header (76μs)
      • Illuminate\Cache\Events\CacheHit (1.88ms)
      • creating: components.footer (81μs)
      • composing: components.footer (92μs)
      • Illuminate\Cache\Events\CacheHit (997μs)
      • Illuminate\Cache\Events\CacheMissed (355μs)
      • Illuminate\Database\Events\StatementPrepared (752μs)
      • Illuminate\Database\Events\QueryExecuted (953μs)
      • eloquent.retrieved: App\Models\SubjectCat (57μs)
      • Illuminate\Cache\Events\KeyWritten (373μs)
      • Illuminate\Cache\Events\CacheHit (237μs)
      • Illuminate\Cache\Events\CacheHit (138μs)
      • Illuminate\Cache\Events\CacheHit (119μs)
      • Illuminate\Cache\Events\CacheHit (200μs)
      • Illuminate\Cache\Events\CacheHit (136μs)
      • Illuminate\Cache\Events\CacheHit (191μs)
      • Illuminate\Cache\Events\CacheHit (421μs)
      • Illuminate\Cache\Events\CacheHit (202μs)
      • Illuminate\Cache\Events\CacheHit (300μs)
      • Illuminate\Cache\Events\CacheHit (178μs)
      • Illuminate\Cache\Events\CacheHit (201μs)
      • Illuminate\Cache\Events\CacheHit (249μs)
      • creating: components.forms.contact-us (306μs)
      • composing: components.forms.contact-us (163μs)
      • creating: components.forms.get-started (195μs)
      • composing: components.forms.get-started (97μs)
      • creating: components.forms.free-tool-download (144μs)
      • composing: components.forms.free-tool-download (74μs)
      • creating: components.forms.claim-free-worksheet (136μs)
      • composing: components.forms.claim-free-worksheet (74μs)
      • creating: components.forms.tutor-subscription-waitlist (163μs)
      • composing: components.forms.tutor-subscription-waitlist (74μs)
      • creating: components.forms.tutor-subscription-join (139μs)
      • composing: components.forms.tutor-subscription-join (74μs)
      • creating: components.forms.tutor-support (134μs)
      • composing: components.forms.tutor-support (75μs)
      • 312 x Illuminate\Cache\Events\CacheHit (19.21%)
        90.74ms
        19 x Illuminate\Database\Events\StatementPrepared (6.18%)
        29.18ms
        19 x Illuminate\Database\Events\QueryExecuted (4.82%)
        22.77ms
        10 x Illuminate\Cache\Events\KeyWritten (1.97%)
        9.32ms
        10 x Illuminate\Cache\Events\CacheMissed (1.96%)
        9.26ms
        1 x Illuminate\Foundation\Events\LocaleUpdated (0.99%)
        4.70ms
        1 x Illuminate\Routing\Events\Routing (0.28%)
        1.33ms
        1 x Illuminate\Database\Events\ConnectionEstablished (0.25%)
        1.18ms
        11 x eloquent.retrieved: App\Models\SubjectCat (0.18%)
        857μs
        1 x Illuminate\Routing\Events\RouteMatched (0.12%)
        563μs
        1 x creating: site.layouts.app (0.1%)
        483μs
        1 x creating: components.canonical (0.09%)
        418μs
        1 x eloquent.booting: App\Models\SubjectCat (0.07%)
        325μs
        1 x creating: site.headers.header (0.07%)
        321μs
        1 x creating: components.forms.contact-us (0.06%)
        306μs
        1 x creating: components.breadcrumbs (0.06%)
        280μs
        7 x eloquent.retrieved: App\Models\HomeworkLibrary\HomeworkLibrary (0.06%)
        276μs
        1 x eloquent.booted: App\Models\HomeworkLibrary\HomeworkLibrary (0.05%)
        228μs
        1 x creating: components.open-graph (0.05%)
        215μs
        1 x eloquent.booting: App\Models\HomeworkLibrary\HomeworkLibrary (0.05%)
        214μs
        4 x eloquent.retrieved: App\Models\HomeworkLibrary\HomeworkLibraryFile (0.04%)
        201μs
        2 x eloquent.retrieved: App\Models\Subject (0.04%)
        197μs
        1 x creating: components.forms.get-started (0.04%)
        195μs
        1 x eloquent.booting: App\Models\HomeworkLibrary\HomeworkLibraryFile (0.04%)
        189μs
        1 x composing: homework.show (0.04%)
        172μs
        1 x creating: components.forms.tutor-subscription-waitlist (0.03%)
        163μs
        1 x composing: components.forms.contact-us (0.03%)
        163μs
        1 x creating: homework.show (0.03%)
        162μs
        1 x creating: components.forms.free-tool-download (0.03%)
        144μs
        1 x creating: components.forms.tutor-subscription-join (0.03%)
        139μs
        1 x creating: components.forms.claim-free-worksheet (0.03%)
        136μs
        1 x creating: components.forms.tutor-support (0.03%)
        134μs
        1 x eloquent.booting: App\Models\Subject (0.03%)
        122μs
        1 x composing: components.breadcrumbs (0.02%)
        116μs
        1 x composing: components.canonical (0.02%)
        102μs
        1 x composing: components.forms.get-started (0.02%)
        97μs
        1 x composing: components.footer (0.02%)
        92μs
        1 x creating: components.footer (0.02%)
        81μs
        1 x composing: components.open-graph (0.02%)
        78μs
        1 x composing: site.headers.header (0.02%)
        76μs
        1 x composing: components.forms.tutor-support (0.02%)
        75μs
        1 x composing: components.forms.free-tool-download (0.02%)
        74μs
        1 x composing: components.forms.tutor-subscription-join (0.02%)
        74μs
        1 x composing: components.forms.claim-free-worksheet (0.02%)
        74μs
        1 x composing: components.forms.tutor-subscription-waitlist (0.02%)
        74μs
        1 x eloquent.booted: App\Models\Subject (0.01%)
        61μs
        1 x eloquent.booted: App\Models\SubjectCat (0.01%)
        60μs
        1 x eloquent.booted: App\Models\HomeworkLibrary\HomeworkLibraryFile (0.01%)
        59μs
        1 x composing: site.layouts.app (0%)
        21μs
      14 templates were rendered
      • 1x homework.showshow.blade.phpblade
      • 1x components.breadcrumbsbreadcrumbs.blade.phpblade
      • 1x site.layouts.appapp.blade.phpblade
      • 1x components.canonicalcanonical.blade.phpblade
      • 1x components.open-graphopen-graph.blade.phpblade
      • 1x site.headers.headerheader.blade.phpblade
      • 1x components.footerfooter.blade.phpblade
      • 1x components.forms.contact-uscontact-us.blade.phpblade
      • 1x components.forms.get-startedget-started.blade.phpblade
      • 1x components.forms.free-tool-downloadfree-tool-download.blade.phpblade
      • 1x components.forms.claim-free-worksheetclaim-free-worksheet.blade.phpblade
      • 1x components.forms.tutor-subscription-waitlisttutor-subscription-waitlist.blade.phpblade
      • 1x components.forms.tutor-subscription-jointutor-subscription-join.blade.phpblade
      • 1x components.forms.tutor-supporttutor-support.blade.phpblade
      uri
      GET college-homework-library/{category}/{subject}/{id}
      middleware
      web, utm.parameters
      controller
      App\Http\Controllers\HomeworkLibraryController@show
      namespace
      where
      as
      homework.show
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      app/Http/Controllers/HomeworkLibraryController.php:79-176
      19 statements were executed, 5 of which were duplicates, 14 unique. Show only duplicated37.78ms
      • Connection Establishedtwenty4_siteHomeworkLibraryController.php#91
        Backtrace
        • 13. app/Http/Controllers/HomeworkLibraryController.php:91
        • 14. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 15. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 16. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
        • 17. vendor/laravel/framework/src/Illuminate/Routing/Route.php:205
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        13.91mstwenty4_siteHomeworkLibraryController.php#97
        Bindings
        • 0: published
        • 1: 0
        • 2: 45665
        Backtrace
        • 16. app/Http/Controllers/HomeworkLibraryController.php:97
        • 17. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 18. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 19. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
        • 20. vendor/laravel/framework/src/Illuminate/Routing/Route.php:205
      • select * from `subjects` where `subjects`.`id` in (1)
        1.17mstwenty4_siteHomeworkLibraryController.php#97
        Backtrace
        • 21. app/Http/Controllers/HomeworkLibraryController.php:97
        • 22. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 23. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 24. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
        • 25. vendor/laravel/framework/src/Illuminate/Routing/Route.php:205
      • select * from `solutionslibrary_files` where `solutionslibrary_files`.`solutionlib_id` in (45665)
        3.98mstwenty4_siteHomeworkLibraryController.php#97
        Backtrace
        • 21. app/Http/Controllers/HomeworkLibraryController.php:97
        • 22. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 23. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 24. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
        • 25. vendor/laravel/framework/src/Illuminate/Routing/Route.php:205
      • select * from `subject_cats` where `subject_cats`.`id` = 1 limit 1
        1.06mstwenty4_siteHomeworkLibrary.php#201
        Bindings
        • 0: 1
        Backtrace
        • 20. app/Models/HomeworkLibrary/HomeworkLibrary.php:201
        • 26. app/Http/Controllers/HomeworkLibraryController.php:105
        • 27. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 28. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 29. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
      • select * from `solutionslibrary` where `id` <> 45665 and `subject` = 1 and `status` = 'published' and `price` > 0 order by RAND() limit 6
        2.79mstwenty4_siteHomeworkLibraryRepository.php#30
        Bindings
        • 0: 45665
        • 1: 1
        • 2: published
        • 3: 0
        Backtrace
        • 14. app/Repositories/HomeworkLibraryRepository.php:30
        • 15. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 16. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 18. app/Repositories/HomeworkLibraryRepository.php:39
        • 19. app/Http/Controllers/HomeworkLibraryController.php:139
      • select * from `subjects` where `subjects`.`id` in (1)
        1.24mstwenty4_siteHomeworkLibraryRepository.php#30
        Backtrace
        • 19. app/Repositories/HomeworkLibraryRepository.php:30
        • 20. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 21. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 23. app/Repositories/HomeworkLibraryRepository.php:39
        • 24. app/Http/Controllers/HomeworkLibraryController.php:139
      • select * from `solutionslibrary_files` where `solutionslibrary_files`.`solutionlib_id` = 45665 and `solutionslibrary_files`.`solutionlib_id` is not null and `publish` = 'question' order by `order` asc, `id` asc
        1.08mstwenty4_siteHomeworkLibrary.php#260
        Bindings
        • 0: 45665
        • 1: question
        Backtrace
        • 15. app/Models/HomeworkLibrary/HomeworkLibrary.php:260
        • 16. app/Transformers/HomeworkLibrary/HomeworkLibraryTransformer.php:58
        • 19. vendor/league/fractal/src/TransformerAbstract.php:128
        • 20. vendor/league/fractal/src/TransformerAbstract.php:107
        • 21. vendor/league/fractal/src/Scope.php:383
      • select * from `solutionslibrary_files` where `solutionslibrary_files`.`solutionlib_id` = 45665 and `solutionslibrary_files`.`solutionlib_id` is not null and `publish` = 'teaser' order by `order` asc, `id` asc
        880μstwenty4_siteHomeworkLibrary.php#260
        Bindings
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        Backtrace
        • 15. app/Models/HomeworkLibrary/HomeworkLibrary.php:260
        • 16. app/Transformers/HomeworkLibrary/HomeworkLibraryTransformer.php:69
        • 19. vendor/league/fractal/src/TransformerAbstract.php:128
        • 20. vendor/league/fractal/src/TransformerAbstract.php:107
        • 21. vendor/league/fractal/src/Scope.php:383
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        880μstwenty4_siteHomeworkLibrary.php#260
        Bindings
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        Backtrace
        • 15. app/Models/HomeworkLibrary/HomeworkLibrary.php:260
        • 16. app/Transformers/HomeworkLibrary/HomeworkLibraryTransformer.php:80
        • 19. vendor/league/fractal/src/TransformerAbstract.php:128
        • 20. vendor/league/fractal/src/TransformerAbstract.php:107
        • 21. vendor/league/fractal/src/Scope.php:383
      • select * from `subject_cats` where `subject_cats`.`id` = 1 limit 1
        930μstwenty4_siteHomeworkLibrary.php#201
        Bindings
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        Backtrace
        • 20. app/Models/HomeworkLibrary/HomeworkLibrary.php:201
        • 32. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
        • 33. vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58
        • 34. vendor/livewire/livewire/src/ComponentConcerns/RendersLivewireComponents.php:69
        • 35. vendor/laravel/framework/src/Illuminate/View/Engines/CompilerEngine.php:70
      • select * from `subject_cats` where `subject_cats`.`id` = 1 limit 1
        1.03mstwenty4_siteSubject.php#100
        Bindings
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        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 3 limit 1
        1.01mstwenty4_siteSubject.php#100
        Bindings
        • 0: 3
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 10 limit 1
        1.38mstwenty4_siteSubject.php#100
        Bindings
        • 0: 10
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 33 limit 1
        1.15mstwenty4_siteSubject.php#100
        Bindings
        • 0: 33
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 11 limit 1
        1.21mstwenty4_siteSubject.php#100
        Bindings
        • 0: 11
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 5 limit 1
        950μstwenty4_siteSubject.php#100
        Bindings
        • 0: 5
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 34 limit 1
        860μstwenty4_siteSubject.php#100
        Bindings
        • 0: 34
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 31 limit 1
        1.19mstwenty4_siteSubject.php#100
        Bindings
        • 0: 31
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 36 limit 1
        1.08mstwenty4_siteSubject.php#100
        Bindings
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        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 28. view::components.footer:170
      App\Models\SubjectCat
      11SubjectCat.php
      App\Models\HomeworkLibrary\HomeworkLibrary
      7HomeworkLibrary.php
      App\Models\HomeworkLibrary\HomeworkLibraryFile
      4HomeworkLibraryFile.php
      App\Models\Subject
      2Subject.php
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          status_text
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          request_cookies
          []
          
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