Classroom connections November 21, 2024

Content ideas and data for math and statistics classes

Classroom connections are less something you can apply right away and more of me pointing out resources. I believe that there is a lot of amazing math, statistics, and data out there that might be used in math classrooms and is also interesting to the curious. At the same time, the next time someone asks what math is used for, well, here are a bunch of examples.

Some of these examples may require some effort to be classroom-ready, and some of this has appeared in previous posts. I’m ordering this roughly based on the level of math. Please let me know if you or someone you know uses any of this in the classroom or has a relevant idea (email me at thomas.pfaff@sustainabilitymath.org). Really, leave a comment (if nothing else, comments are encouragement for me to keep doing this) or send me an email if these classroom connection posts are useful. Similarly, if you have an opinion or concept that is relevant, please share it in the comments. Please share, like, subscribe, and feel free to comment.

Data, graphs, modeling —population pyramids

Population Momentum and Fertility (Jan 2024, pdf file) is a report from the U.S. Census Bureau and has numerous population pyramids under different scenarios. These are excellent for talking about population modeling.

The paper also has the math for population momentum. Here is a peak.

An abundance of confidence intervals and p-values

If you just want to give students confidence intervals or p-values to interpret, then check out Sex Differences in Cardiovascular Outcomes in Patients With Kidney Failure (5/3/2024). Here is just one example. This paper was highlighted in QTRS 11/7/2024.

Good statistics study and ideas for project

The paper Familiar Sequences Are Processed Faster Than Unfamiliar Sequences, Even When They Do Not Match the Count-List (7/9/2024) is a nice understandable study. The results are interesting to math, and the paper could be used in a statistics class either to talk about or to generate ideas for a project. This was highlighted in QTRS 8/1/2024.

In order processing, consecutive sequences (e.g., 1-2-3) are generally processed faster than nonconsecutive sequences (e.g., 1-3-5) (also referred to as the reverse distance effect). A common explanation for this effect is that order processing operates via a memory-based associative mechanism whereby consecutive sequences are processed faster because they are more familiar and thus more easily retrieved from memory. Conflicting with this proposal, however, is the finding that this effect is often absent. A possible explanation for these absences is that familiarity may vary both within and across sequence types; therefore, not all consecutive sequences are necessarily more familiar than all nonconsecutive sequences.

Useful data

It seems that the data here, Global change and premature hatching of aquatic embryos (9/5/2024), could be useful for more basic statistical tests if not for the models they did. They provide links to the data at the bottom.

Data from (a) 37 unhatched stickleback embryos, (b) 102 unhatched black goby embryos and (c) 7 unhatched pumpkinseed sunfish embryos were used to plot embryo activity curves (smoothed black GAM curve, fitted to the averaged movement of all unhatched embryos within a trial at each temperature point) and calculate the mean critical thermal maximum (CT_max) ± SE (vertical dashed lines and shaded bars) for all species (data from unhatched stickleback and black goby embryos are published in Cowan et al., 2023). CT_max was taken as the last movement of each individual embryo. Also presented (in red symbols) are data for embryos that hatched prematurely during the trial, and smoothed red GAM curves have been fitted to the movement of these individuals until the point of hatching (N_stickleback = 11; N_{black goby} = 10, N_pumpkinseed = 36).

Regression

There are links to the data in the paper Normalized difference vegetation index analysis reveals increase of biomass production and stability during the conversion from conventional to organic farming (8/28/2024), so you could reproduce all of these regressions or just ask students to interpret them.

Relationships between latitude and soil properties, and NDVI (left panel) and its temporal stability (right panel). Regression lines indicate significant associations. Red dots = conventional management, blue dots = organic management. NDVI, normalized difference vegetation index.

Hierarchical clustering and more

Global biogeographic regions for ants have complex relationships with those for plants and tetrapods (7/5/2024) has links to the data sources, and so theoretically one could reproduce this graph or at least (d). As a bonus, there is a heat map graph below.

Ant biogeographic realms and regions of the world are shown in a, and the associated dendrogram resulting from hierarchical clustering based on phylogenetic turnover (Pβsim) across 5 × 10 ⁴km² hexagons is shown in b. Ordination results using nonmetric multidimensional scaling (NMDS) show relationships among realms in c and regions in d. Two newly defined species-level realms and their affiliated regions are underlined. Black solid lines and white lines (dashed on the ocean) delineate the borders of realms and regions, respectively. Colors used to characterize particular realms in maps correspond to those in dendrogram and ordination plots. Areas without sufficient data or invalid biogeographic units are indicated in gray. The map in a is in the Robinson projection system, and the dendrogram in b used the unweighted pair-group method using arithmetic average (UPGMA).

Data science Stuff

There is a lot going on in Exploring Early Number Abilities With Multimodal Transformers (9/3/2024). Maybe good for an independent study or just an example of applications. Here is one graph to get you interested in the paper.

Hierarchically clustered heatmap of the cosine similarity between aggregated activation trajectories for each task in the dataset.

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Comments

Please point out if you think something was expressed wrongly or misinterpreted. I'd rather know the truth and understand the world than be correct. I welcome comments and disagreement. We should all be forced to express our opinions and change our minds, but we should also know how to respectfully disagree and move on. Send me article ideas, feedback, or other thoughts at briefedbydata@substack.com.

Bio

I am a tenured mathematics professor at Ithaca College (PhD Math: Stochastic Processes, MS Applied Statistics, MS Math, BS Math, BS Exercise Science), and I consider myself an accidental academic (opinions are my own). I'm a gardener, drummer, rower, runner, inline skater, 46er, and R user. I’ve written the textbooks R for College Mathematics and Statistics and Applied Calculus with R. I welcome any collaborations.