I see that you've got all these semantic and interactive mathematical ideas and I don't know it's similar to being very good at being an academic in that sense. I got this representation of mathematics and it's all semantic. So say that I'm seeking to emphasize the importance of identity and engagement in our learning..and seek I did. What we're doing is we've got this strategy to make mathematics more engaging and accessible for all of us and we've got quite a diverse range of students who have the desire to reach metamathematical space and learn about the lens of chemistry and how we start out with these molecules that come in, and then those atoms will be the atoms that come out and then we can learn about all kinds of things whether it's name reactions or SMILES strings. So what do we have to say about equity in mathematics education? Suppose that we want to incorporate some interactive elements of metamathematical space but we can't. But we can incorporate elements that allow the manipulation of certain variables and that, might increase student engagement significantly.
Manipulate[
Plot[m*x + b,
{x, -10, 10},
PlotRange -> {{-10, 10}, {-20, 20}},
AxesLabel -> {"x", "y"},
PlotLabel -> "y = mâ‹…x + b"],
{{m, 1, "Slope (m)"}, -10, 10, 0.5},
{{b, 0, "y Intercept (b)"}, -10, 10, 0.5}
]
It's sort of like when we learned the periodic table, of the elements, and we grind through and do that by the industrial methods of solving these equations; just because the reaction balances - the equation balances - doesn't mean that the equation will actually happen. The periodic table exists independently of the equations. All this equation solving is like drinking from a hose, and if you question me further I would say that there are a lot of things that just get continually updated..and so even if we try to talk about the different meanings of metamathematical space I would say that it doesn't end when we simulate the impact of teaching practices on student engagement and mathematical identity over a number of steps. It's not the kind of visual representation of how different teaching interests - Problem Solving, Discussion-Centered, Conceptual Understanding - affect the identity trajectories and engagement of students which is something I definitely strive for. It's also and really about the sinusoidal functions, you know how the amplitude-frequency-phase shift paradigm can just be adjusted to see the effect that the sine wave, which exists outside of the paradigm, can have.
About 130 years ago, we used to do stamp collecting or physics whatever you call it, you could even have a discussion on slope-intercept-based pedagogy in these graphs, and by adjusting the inputs and observing the outputs in each simulation, educators like us can actively engage with the model. And a lot of these things you won't find in any textbook. Changes in teaching style whether it's mirroring the active learning and problem-solving aspect of the Wolfram teaching practices..and I'd like to believe that there's some metamathematical meaning to this but the more we change over time in response to teaching practices the more informed decisions in teaching methods we'll get, and now for the focus areas you've been developing...how can students' mathematical identities and engagement levels be visualized over a set number of classroom interactions? With some long-chain trajectories we can show how each practice impacts engagement and identity over time, which visualizes the complex interplay between teaching methods and student outcomes.
SeedRandom[1231231234];
teachingInterests = {
"ProblemSolving",
"DiscussionCentered",
"ConceptualUnderstanding"};
studentEngagement = 0.5;
mathematicalIdentity = 0.5;
ClassroomDynamics[steps_] := Module[
{engagementImpact,
identityImpact,
engagementHistory,
identityHistory},
{engagementImpact,
identityImpact} = {studentEngagement,
mathematicalIdentity};
engagementHistory = {engagementImpact};
identityHistory = {identityImpact};
Do[engagementImpact += RandomReal[{-0.15, 0.05}];
identityImpact += RandomReal[{-0.05, 0.15}];
engagementImpact = Min[Max[engagementImpact, 0], 1];
identityImpact = Min[Max[identityImpact, 0], 1];
AppendTo[engagementHistory, engagementImpact];
AppendTo[identityHistory, identityImpact];
, {steps}];
{engagementHistory, identityHistory}];
StudentBloomWalkVisual[practices_, steps_] := Module[
{engagementHistories, identityHistories, colors, plots},
{engagementHistories, identityHistories} =
Transpose[ClassroomDynamics[#, steps] & /@ practices];
colors = ColorData[97, "ColorList"][[1 ;; Length[practices]]];
bigPicture = MapThread[
ListLinePlot[
Transpose[{#1, #2}],
Mesh -> Full,
MeshStyle -> PointSize[Medium],
PlotStyle -> #3,
AxesLabel -> {"Engagement", "Identity"},
PlotRange -> {{0, 1}, {0, 1}},
GridLines -> Automatic,
Frame -> True,
ImageSize -> Large,
PlotRangePadding -> Scaled[0.1],
AspectRatio -> 1/GoldenRatio,
ImagePadding -> {{50, 200}, {50, 50}},
PlotLabels -> Callout[#4, Automatic,
LabelStyle -> {FontSize -> 12, FontColor -> #3},
LeaderSize -> {Automatic, Automatic, 20}]] &,
{engagementHistories, identityHistories, colors, practices}];
Show[bigPicture, ImageSize -> 800]];
StudentBloomWalkVisual[teachingInterests, 30]
This random walk might passively demonstrate how we can make visual representations more tangible and support discussion-centered learning. What if we want to add some table of values or discuss the relationship between the values and the graph, which would make things more text-based and provide a softer landing for those of us who don't just want to draw sticks but also want to build a conceptual understanding of how these vectorized parameters affect the shape and position of trigonometric functions? We could provide a real, tangible example of its own and actually create an environment where students can better understand and discuss mathematical concepts, by manipulating mathematical functions through code.
Manipulate[
Module[
{f, xVals, yVals, mustardSeedsSynthesis, researchTables,
pedagogTitles, partitionedData, tableAmalgamate},
f[x_] := a Sin[b x + c];
xVals = Range[-2 Pi, 2 Pi, 0.1];
yVals = f /@ xVals;
pedagogTitles = {
Style["x", Bold, 7,
Background -> Lighter[RGBColor[1, 215/255, 0], 0.9]],
Style["f(x)", Bold, 7]};
partitionSize = Ceiling[Length[xVals]/6];
partitionedData = Partition[
Transpose[{xVals, yVals}],
partitionSize];
mustardSeedsSynthesis = Map[Style[#, FontSize -> 7] &,
#, {-1}] & /@ partitionedData;
researchTables = Table[
Grid[
If[i == 1,
Prepend[mustardSeedsSynthesis[[i]], pedagogTitles],
mustardSeedsSynthesis[[i]]],
Frame -> All,
Background -> {None,
None, {1 -> Lighter[RGBColor[1, 215/255, 0], 0.9]}},
Dividers -> {False, {True, True, {False}, True}}],
{i, Length[mustardSeedsSynthesis]}];
tableAmalgamate = Grid[{researchTables}, Spacings -> {2, 0}];
Labeled[Row[{
Plot[f[x],
{x, -2 Pi, 2 Pi},
PlotStyle -> {Thick, RGBColor[1, 215/255, 0], Dashing[Small]},
PlotRange -> {-10, 10},
AxesLabel -> {"x", "f(x)"},
ImageSize -> 600, Mesh -> Full, MeshStyle -> Purple,
Epilog -> {
Inset[
Style[
Text[{"Amplitude: ", a},
"Frequency: ", b,
"Phase Shift: ", c],
Small, White
],
{Right, Top},
{Right, Top}]},
Background -> Transparent],
Spacer[10], tableAmalgamate}],
"Sinusoidal Computation", Top]],
{{a, 1.4, "Amplitude"}, 0, 5, 0.5, Appearance -> "Labeled"},
{{b, 0.3, "Frequency"}, 0, 5, 0.5, Appearance -> "Labeled"},
{{c, 0, "Phase Shift"}, 0, 2 Pi, 0.1, Appearance -> "Labeled"},
SaveDefinitions -> True
]
This structured approach to presenting data is a catalyst for making it more digestible for students to develop their mathematical identity by manipulating mathematical functions through code; seeing the immediate impact of their actions on the outcomes, in a beautiful golden grid that suggests a structured approach to presenting data..the principles of these embodied principles provide the interactive and conceptually rich environment that engages students and that's a big, strong part of fostering a mathematical culture in the classroom. I remember when Bluetooth first came out and we all ate our pride in order to illustrate the conceptual understanding of the scattered information landscape, and how it's affected by such a large waveform..I wonder how we can improve the click-through rate of these code snippets in order to more fully understand the relationship if it even exists between the quadratic equation and its graph.
Manipulate[
Module[
{f, xValsNegative, xValsPositive, yValsNegative, yValsPositive,
tableNegative, tablePositive, amazingTableAmalgamate, forwardShare},
f[x_] := a x^2 + b x + c;
xValsNegative = Range[-9.5, 0, 0.5];
xValsPositive = Range[0.5, 10, 0.5];
yValsNegative = f /@ xValsNegative;
yValsPositive = f /@ xValsPositive;
titleForwardShare = {Style["x", Bold, 7,
Background -> Lighter[RGBColor[1, 215/255, 0], 0.9]],
Style["f(x)", Bold, 7]};
tableNegative = Grid[
Prepend[
Transpose[
{Style[#, 7] & /@ xValsNegative,
Style[#, 7] & /@ yValsNegative}],
titleForwardShare],
Frame -> All,
Background -> {None,
None, {1 -> Lighter[RGBColor[1, 215/255, 0], 0.9]}},
Dividers -> {False, {True, True, {False}, True}}];
tablePositive = Grid[
Prepend[
Transpose[
{Style[#, 7] & /@ xValsPositive,
Style[#, 7] & /@ yValsPositive}],
titleForwardShare],
Frame -> All,
Background -> {None,
None, {1 -> Lighter[RGBColor[1, 215/255, 0], 0.9]}}
, Dividers -> {False, {True, True, {False}, True}}];
amazingTableAmalgamate =
Row[{tableNegative, Spacer[20], tablePositive}];
Row[{Plot[f[x],
{x, -10, 10},
PlotStyle -> {Thick, Green},
PlotRange -> {{-10, 10}, {-10, 10}},
AxesLabel -> {"x", "f(x)"},
PlotLabel -> "Interactive Quadratic Rewriting",
ImageSize -> Large],
Spacer[10],
amazingTableAmalgamate}]],
{{a, 1, "Coefficient a"}, -5, 5, 0.1, Appearance -> "Labeled"},
{{b, 1, "Coefficient b"}, -5, 5, 0.1, Appearance -> "Labeled"},
{{c, 0, "Coefficient c"}, -5, 5, 0.1, Appearance -> "Labeled"},
SaveDefinitions -> True
]
How is the transmission rate of various diseases correlated with the fundamental concept of slope and intercept in linear & quadratic equations? Or, how can we compartmentalize all of the characteristics of a function into a visual demonstration such that whenever there's an elephant in the room like, let's say a student omits some problems...how can we talk about..mathematical concepts and take a deep dive because I do, I use the concept of slope and intercept, titration and derivatives in linear equations it's my all-time favorite way to show that mathematics is seen as a dynamic and applicable field rather than a static set of rules and procedures. Relating mathematical concepts to real-world scenarios shows how we can reason and solve problems in a way of thinking, in an organized fashion about the world, whether it's in the context of thinking about infectious to susceptible, the meaning of mathematical representation of Salmonella or cross-sensory designation of scenery that pops, where students' perceptions of themselves as learners and practitioners of mathematics can significantly impact their academic performance..we want to show them what we've got, because anybody who's been in mathematics or STEM-related fields would have a pedagogical understanding of how we can reduce passive learning and computationally engage students in applying mathematical concepts through the creative approach which fosters positive mathematical identities. And I use it, I use Mathematica all the time, the interconnected isomorphisms of the Ruliad have always been a part of my positive self-concept.
