I have some forthcoming studies on this as well as past related publications, but here are a few ways: helping students to see that they can use math to build their mind; helping students see that much of nature is ordered mathematically (e.g., foot size and the orbit of the planets around the sun); helping students understand how the rigorous and methodical type of thinking involved in math will help with everyday problem-solving; helping students understand how math plays a role in many of the careers they might be interested in (e.g., technology, health, management, and engineering); otherwise helping them see how math is intriguing or can help them meet psychological needs. The strongest predictor of engagement in academics is often intrinsic motivation (e.g., see Froiland & Worrell, 2016), so all of the teacher autonomy support research is relevant (e.g., see Su & Reeve, 2011).  

Article Intrinsic motivation, learning goals, engagement, and achiev...

Thank you for adding your ideas and sharing your research, John. I look forward to reading the article.

Generally for older (adult) students, having them fully appreciate the utility of the mathematical concepts is key.  For instance, teaching nursing students math concepts (ratios, elapsed time, etc.) is crucial to patient care.  Using case studies and short cases that have high fidelity to actual professional encounters is helpful.

Teachers can certainly make engaging early math activities, like crafts to find patterns.  Search the web for math activities by grade and you'll find an incredible range of cute examples.  I don't feel it's particularly difficult to design engaging activities for math at higher levels.  There are many analogous activities for older students.  For example, I've had students make their own slide rules from paper.  Seeing how exponents allow simply sliding paper to multiply astounds even high schoolers.  I've taught high school physics and students love experimenting with the everyday world.  In physics class they apply math they learned in prior school years, but there is no reason you couldn't repurpose simple physics studies to teach the math.  For example, set up a shallow ramp and roll down a ball repeatedly.  Make XY pairs for time and distance.  Plot the graph and calculate the best fit function.  Students discover that the distance travelled is a quadratic function of the time.  Best wishes engaging your students Barbara! ~ Kevin

Thanks to Claire and Kevin for the additions, and I certainly hope there will be more. I don't teach math, but am passing along the responses to a colleague working on a graduate school project.

I use physical models to explain abstract concepts.

In a problem where (say) 3 times the son's age exceeds twice his father's age by 9.

I use a weighing scale to describe this: on one side we have 3 times the son's age.

on the other side, we have 2 times the father's age.

Since the son's age side is heavier, the scale will tilt on that side. In order to balance the scale, I have to minus from the son's side or add on the father's side.

Thus I get the equation: 3 times the son's age - 9 = twice the father's age

or 3 times the son's age = twice the father's age + 9

As an outsider, (I teach physics at the university level) I have an observation which might be useful.  I use math as a language to communicate relationships I already know something about more clearly.  In this language analogy, I help students write paragraphs and short stories.  Their math class might easily seem just an old boring "grammar class," in this analogy. 

I feel like the most effective way to connect with students in math is to use it to tell a story that they know, and preferably care about.  If the math simply says something that they already know, then the new math concept is more meaningful.  I explicitly explain this to students early in the courses, and whenever it pops up.

In practice, students, though they regularly forget some math, no longer complain about the math in my physics courses.  They claim that it "isn't that bad."

In my sections of a General Education course called Light, Sound, and Perception, we use logs for sound level calculations.  Though they are allowed to use calculators, I regularly get students who dislike math estimating logs for sound levels without a calculator!  While it is only two or three out of 60, getting any converts sounds like a huge success.

I am told that according to some study, that learning math without context, though more painful, makes it more transferable. I don't know about that. If so, generalizing the concepts after seeing the relationships might meet the need.   I suspect that it might also help if students designed, executed and presented to each other projects in which they see the value of the math.

Often students don't see the connections between what they learn in math, and what they use in other settings.  Here is another example. 

Electricity and Magnetism requires multi-variable calculus at least concurrently.  I made a fuss about drawing "little bits" of length, area, and volume accurately described when doing integrals.  A student later told me that on his calculus exam, "I totally froze.  So I tried your' little bits.'  THAT WORKS TOO!" 

He was completely unaware that I was simply trying to do exactly what his math class did.  He thought that they were entirely different techniques with no relationship to each other.

Hello

I argue that the Mathematical Modeling can be a method to succeed in that desire. We have developed several modeling practices, as a template can be adapted to different levels of education.

Esero have contributed.