Category: General Discussion

This might sound counterintuitive coming from a company that has literally hundreds of online math lessons available on YouTube, but for math to be learned well, students must practice on paper. 

Study after study has shown as much:

– A 2020 study by the Reboot Foundation found that of students using an online math tool, those who were encouraged to do their calculations using pencil and paper did an average of 13 points better than those who weren’t. 

– A 2014 series of studies published in Psychological Science found that those students who take notes by hand (which allows them to summarize, paraphrase, and concept map) perform better than students who type notes quickly verbatim on a laptop. 

– A 2017 review of research done since 1992 found that students were able to better comprehend information in print for texts that were more than a page in length, which is apparently related to the disruptive effect that scrolling has on comprehension.

– The same authors as the 2017 review conducted their own three studies on college students and found that comprehension was better for print reading than it was for digital reading. 

– A 2021 Tokyo study shows stronger brain activity after writing on paper than on a tablet or smartphone, suggesting that unique, complex information in analog methods likely gives the brain more details to trigger memory. 

And let’s not forget the importance of “showing your work” to help students grasp concepts and avoid guessing while allowing teachers to troubleshoot incomprehension and pinpoint mistakes in calculatory processes. 

All of the above is why eMATHinstruction sells workbooks to school districts and offers free print-outs to teachers — so that students can work out problems on paper. 

All this is not to say that there aren’t effective uses for digital technology when it comes to teaching and learning math. For example, the interactivity and dynamics of digital tools can be particularly helpful in geometry education, where comprehending visual and dynamic geometrical objects and relations is necessary. And tech certainly has its benefits: with more and more communities gaining access to internet infrastructure, kids can use and reuse infinite online learning tools, making learning more equitable; education can happen across borders; online files can’t be damaged the way physical textbooks can; and relying on less paper is more sustainable. The digital tools currently available haven’t yet reached their full potential: the features of digital resources — e.g. dynamics, feedback, personalization, and cooperation — can and should be improved upon. The sky’s the limit! Or should we say, the potential is unbounded by any finite number.

As with most things in life, a balanced approach is best: use and improve upon technology to aid in the teaching and learning of math, but don’t give up on the tried-and-true method of pencil on paper. Good Will Hunting would not have won an Oscar had Matt Damon done his calculations on a computer instead of a chalkboard. Plus, pencil sharpeners are just plain fun!

Happy autumn to all our eMATH teachers, students, and parents out there. The school year is now well underway around the country. Here’s what’s new and need-to-know from eMATH this October 2022!

OCTOBER MATH CONFERENCES

We are gearing up for these upcoming conferences — hope to see you at one of them! Remember to take a selfie in front of our eMATH booth and post to social media with #eMATHinstruction for a chance to win a big eMATH Fun Box! And be sure to tell your math-teacher friends, who aren’t yet familiar with how eMATHinstruction.com can make teaching easier, to stop by our booth: we’ll have lots of eMATH swag and info!

1. The Association of Math Teachers of New Jersey
OCT 20
Plainsboro Township, NJ

2. The Mississippi Council of Teachers of Mathematics
OCT 21-22
Flowood, MS

3. The Association of Math Teachers of New York State
OCT 28-29
Rochester, NY

NEW MATERIALS FOR TEACHERS

Here are the new teacher tools to use in the classroom for each of our courses:

N-Gen Math 6:

• Unit 3 (Decimals) Form B unit assessment
• Unit 4 (Negative Numbers) practice worksheet on points in the coordinate plane
• Unit 4 Form B unit assessment

N-Gen Math 7:

• Practice set of problems in Unit 4 (Percent) dealing with percentage word problems.
• Unit 4 Form B assessment
• Practice set of problems from Unit 5 (Linear Expressions) on like terms, recognizing, and combining

N-Gen Math 8:

• Unit 3 (Transformations) practice set of problems on transforming figures in the coordinate plane
• Unit 3 Form B unit assessment

N-Gen Math Algebra I:

• Unit 4 (Linear Functions) Exit Tickets.
• Unit 4 Form A unit assessment
• Unit 4 modeling problems on modeling the boundaries of a shed using linear functions as well as horizontal and vertical lines

Common Core Geometry:

• Unit 3 (Euclidean Geometry Proof) Form D mid-unit quiz
• Unit 3 practice set of problems involving proofs that have partitioning in them (some of these proofs are shorter partition proofs and some are longer proofs that simply have partitioning in them)

Common Core Algebra II:

• Unit 4 (Exponential and Logarithmic Functions) Form D mid-unit quiz
• Practice set of problems from Unit 4 on creating the equations of exponential functions given two points on the function
• Unit 5 (Sequences and Series) Form D mid-unit quiz

Enjoy the fall foliage and all the colorful fractals — science says it will actually chill you out!

Every state has its own state bird, its own culinary stand-outs, its own unique personality, and its own math standards. So how can teachers be sure our eMATHinstruction materials meet their state’s math standards? By using our new Standards Alignment Tool! 

eMATHinstruction curricula were originally designed to align to the Common Core State Standards (CCSS).  Our new N-Gen Math series is designed around the Next Generation Mathematics Learning Standards (NGMLS), which are very similar to the CCSS.  In fact, most state standards are closely aligned to the CCSS with slight tweaks to certain topics. By using our Standards Alignment Tool (which is powered by EdGate), you can see which lessons meet which standards in your particular state! 

Please note: The N-Gen Math Algebra I alignment is expected by November 2022.

10 Quick Steps to Using Our Standards Alignment Tool 

1. Log in to your account. (FYI: some browsers work better than others; EdGate recommends that you use Firefox.)

2. Either click on “FOR TEACHERS” in the nav bar, select the course you’re interested in (for example, N-Gen Math 7 or Common Core Algebra I or…), and click on the blue tab that reads “Standards Alignment – Powered by EdGate” (it’s the first link under the description of the course)…

3. OR skip Step 2 and simply go to the main Standards Alignment page at emathinstruction.com/standards-alignment. You can also find a link to it at the bottom of the pulldown menu under “FOR TEACHERS” in the nav bar.

4. Decide whether you’d like to do a “Standards” search or a “Content” search. In other words, if you’re looking for resources that meet a specific standard, do a Standards search; if you’re looking for the standards in a specific resource, do a Content search. You can toggle between the two kinds of searches by clicking either one of those blue tabs on top of the search box. 

5. If you do a Standards search, select the relevant options from the various pull-down menus. (For example, a 6th-grade math teacher in California would choose that state, grade 6, and the N-Gen Math 6 course from the pulldowns). Then click the blue “Search” button. You’ll receive a complete list of all the specific standards in your state, each with a “Correlations” button that reveals a complete, clickable list of all the eMATHinstruction lessons, homework, and videos that meet the specific standard. 

6. If you do a Content search, select the relevant options from the various pull-down menus. (For example, a 6th-grade math teacher in California would choose the N-Gen Math 6 course, grade 6, and that state from the pulldowns.) Then click the blue “Search” button. You’ll receive a complete, clickable list of all the relevant eMATHinstruction lessons, homework, and videos with a button to view all the state’s standards that each individual eMATH material meets. 

7. For both types of searches — Standards and Content — you can do an advanced search by entering a specific term to narrow down your results. For example, if you were looking for everything related to fractions that meets your state’s standards for a specific grade, you would select all the relevant options from the pulldowns and then click the “Advanced Options” link and enter the term “fractions” in the text box before clicking the “Search” button. 

8. You can download your results (in either Excel, Word, or PDF form) OR you can print your results by clicking one of the four respective icons at the top right of your search results.

9. Then you can happily use all the eMATHinstruction materials that came up in your search, confident in the knowledge that they align with your state’s standards. 

10. If you can’t find what you are looking for, we are happy to provide a custom report for you.  Please send an email to info@emathinstruction.com with your request.

Here’s a how-to video to walk you through these steps (for optimum visuals, watch full screen):

The debate about the role of homework in math education has been around for quite a long time. How much homework should be assigned? Should homework be collected and graded for accuracy or should it only receive a completion grade. What portion of a student’s grade should depend on homework rather than assessments and projects? There are many in the world of math education who believe that homework should either be deemphasized, not used for grading purposes, or eliminated entirely. Regardless of where you fall on this issue, as educators, we now face the rise of technology that makes it exceptionally easy for students to copy homework solutions without doing the important thinking required to solve them.

I founded eMATHinstruction in 2008.  Over the years we have seen varying degrees of students’ attempts at cheating.  The most blatant cheating by students comes in the form of kids stealing their parent’s credit card to purchase answer key membership on our website.  One of the limitations of us making our keys available to teachers who purchase memberships for themselves is that they are available to anyone with a credit card.  We have learned along the way and have put measures in place to limit access to certain resources to verified teachers.  When one makes a purchase on our website we state clearly that we take academic integrity seriously and will work with school districts who suspect a student of cheating.  This comes up as soon as you add a Teacher Plus membership to your cart.

As with any digital resource it’s sometimes difficult to keep the genie in the bottle.  Just ask any musician who battled Napster.  For print resources one only needs a camera or digital file and the ability to upload (sound like most of America?) for resources to get out there.  Obviously, students have always had the ability to cheat by copying a classmate’s solutions, but when they have the ability to use a search engine to find the answers, it becomes a much larger issue, especially when homework is an integral part of their grade.  This is a new reality for schools and teachers to come to terms with.

From a publisher’s perspective it is frustrating that resources pop up in places without permission.  We try to make it clear that posting of resources is not permissible.  For those of you who use our resources on a regular basis this is a familiar box you see and must accept with every download.

We do what we can on the front end to discourage file uploads but since the beginning of time some people will feel pressured to cheat because they struggle or because it’s the path of least resistance.  Today’s technology makes cheating that much easier, and the pandemic has poured fuel on that fire.

Entire website services have emerged for students to get “help with homework”. Quotes there were intentional. Each one of these sites has an academic integrity policy that states their service is not to be used for cheating.  You would be hiding under a rock if you did not think the primary use of these sites was cheating.  They encourage students to upload the contents of their backpack to get “help”.  That help then remains up for all to see.  One would think this is copyright infringement, and it is, but sites like these hide behind the DMCA.

The Digital Millennium Copyright Act (or the DMCA for short), signed into law in the late 1990’s by Bill Clinton, did two things: (1) it increased penalties for copyright violations on the internet and (2) it limited the liability of online service providers if their users violated copyright laws.  When requested by the copyright holder that material be taken down, the hosting site must comply.  We have become quite familiar with submitting DMCA takedown notices, but it is a game of Whack-a-Mole. 

Obviously, the ability for a student to be able to find solutions to assignments is problematic when it comes to homework. At least this forced students to do a search online. Technology is moving so fast that search engines to find uploads may soon be a thing of the past.  The new king is instant answers from apps such as Photomath and websites like Chegg.  These tools make it even easier for students to cheat on homework and possibly even assessments (quizzes and tests).

If you hadn’t heard of Photomath yet, you need to quickly learn about it, especially if you put a great value on homework in your grading system. Photomath is a free app (with in app purchases) that will solve many math problems, giving step by step solutions to the problem. Photomath markets itself as an app that encourages learning, and in a perfect world, that might be the case. The mere fact that it is able to instantaneously solve a math problem immediately lends itself to cheating. I wanted to see how powerful Photomath was so I created a problem that I thought would be challenging:

Anyone who has taken Algebra can factor a quadratic, especially with a leading coefficient of one. But many students, even those who may have taken precalculus, will not be able to factor a random cubic polynomial. To solve this using Photomath I literally opened the app, framed the problem in the viewing window:

Within the time of a heartbeat, the app produced an answer with a step by step solution:

This isn’t the method I would have thought to use (although I’m fascinated to know if any factorable cubic could be done this way). Still, regardless of how I would have done it, it certainly is an acceptable solution, if perhaps a non-routine one.

In case you’re wondering, the app itself is free and to find the solution to my example problem I did not have to pay anything or sign up for any account. Think about how easy it would be for a student to use this app to do an assignment that is primarily rote fluency problems. I did, however, find that Photomath had a hard time with many of the problems in our Common Core Algebra I book. Any type of problem that had any significant text in it was an issue and multiple-choice problems also seemed to throw the app for a loop. I was surprised by how many times I saw this pop up on the app:

And geometry proofs? Forget about it! Though they do seem rather convinced they will be able to handle them eventually:

This app can definitely crunch through a lot of drill problems. Give your students a 50 problem factoring worksheet for homework and I guarantee it can be done in five minutes by this app. But, what’s a kid to do if this app can’t handle a multiple choice problem as easy as this?

That’s where Chegg comes in. See, the problem with Photomath is that it relies on pretty impressive, yet limited, artificial intelligence to solve the problems. Chegg, on the other hand, employs an army of real people who just wait to solve whatever problem you throw their way. Because Chegg can handle a far greater sophistication of problems, it is the choice of college students. The web service is so popular amongst college students that it has become a verb. Students will “chegg” an assignment if they don’t have the time or ability to do it otherwise.

So, what do you get for your $14.95 per month on Chegg?

Well, if you supply them with a math question you first are given a selection of similar ones that they’ve already worked out solutions for. If you can’t find yours in that selection you can opt to get a handwritten response back. Remember our original problem we gave to Photomath?

The solution Chegg sent me back was this:

This is actually the way I would expect a student of mine in precalculus to factor this cubic polynomial. So, if I saw this solution come back, I would likely give the student full credit (although I do believe when the solver divides the cubic by the linear binomial there should be more work shown to produce the quadratic expression). At this low, low, low monthly price, a student could spend $150 for a 10 month school year and be supplied with the answer to any math problem they want, including those at the college level.

This trend is troubling enough that teachers are certainly taking notice at both the K-12 level and the collegiate level. It was exacerbated this year by remote and hybrid instruction forced by the pandemic. I know many teachers have had no choice but to give assessments to students, even online when they are at home. Perhaps there are ways for the teacher to monitor the use of these types of apps while the student is taking their test, but it isn’t very realistic if they are taking the test at home.

At some point, though, we will return to in school learning everywhere. In the case of tests, then, students will typically not be able to use Photomath or Chegg to help them (unless their teacher is at an almost Hollywood level of cluelessness in the classroom). But, students will certainly be able to continue to use these types of apps at home in order to do their homework. As much as we might wish it, these apps are certainly not going to go away. In fact, now that companies know that they can make a profit from helping students cheat, they are starting to proliferate. Watch for these exciting apps to be showing up on your students’ phones: Slader, Mathway, and Mathpresso. The presence of these types of services must make us ask very fundamental questions about the role of homework, especially in terms of assigning grades to students.

In my own classes, which ranged from Algebra 1 through AP Calculus, I always assigned homework and I always looked it over briefly at the beginning of each class. I hardly ever collected it and it comprised approximately 10 to 15% of the overall grade for any given marking period. When I checked student homework it was mainly for completion and effort. Kids who both completed and made good effort got a 2/2, those who maybe didn’t do all of it or made less effort received a 1/2, and those who didn’t do it received a 0/2. Homework wouldn’t typically make or break a kid, but getting a perfect homework grade would typically raise a student’s grade around 6 to 10%.

We really must ask ourselves what the role of homework should be moving forward and what it should look like given a student’s ability to have much of it done for them (even if it is at a cost). I am not an advocate of no homework at all. That would be as silly as expecting students to learn music by only attending the lesson and never practicing at home. I am an advocate of two things: (1) high quality homework that includes not only fluency problems but application and reasoning tasks and (2) optimizing the weight of the homework grade relative to the student’s marking period grade.

I also want to throw one more idea about homework out there to think about. I taught at the high school level for almost two decades. In my time I saw thousands upon thousands of sets of homework solutions. I would say that a good percentage of the time my two biggest issues with students doing homework were: (1) their lack of doing it at all and (2) the exceptionally poor quality of work when it was done. In both cases, these students would often go on to do poorly on assessments as well (shocker!). I wonder if these students would have actually gained more from copying down perfect solutions that were produced by an app. Does the transfer of a solution from a screen to a piece of paper also transfer some educational knowledge? Do the apps step by step directions actually help students who faithfully copy the solutions? We all cringe at the thought of students doing this and claiming the solutions they copy down as their own. I think we also all understand that the student who wants to learn and who has the fortunate circumstances to learn gets far more from struggling on the homework and producing their own solutions. But, maybe, just maybe, for the students who would otherwise not do any homework or the student who would do the homework, but poorly, these apps might even help.

As always, we would love to have input on how teachers see the role of homework in light of apps like Photomath and services like Chegg.