According to how Silverlight TwoWay binding works, when I change the data in the FirstName field, it should change the value in CheckFirstName field.

Why is this not the case?

ANSWER:

Thank you Jeff, that was it, for others: here is the full solution with downloadable code.

XAML:

<StackPanel>
    <Grid x:Name="GridCustomerDetails">
        <Grid.RowDefinitions>
            <RowDefinition Height="Auto"/>
            <RowDefinition Height="Auto"/>
            <RowDefinition Height="*"/>
        </Grid.RowDefinitions>
        <Grid.ColumnDefinitions>
            <ColumnDefinition Width="Auto"/>
            <ColumnDefinition Width="300"/>
        </Grid.ColumnDefinitions>

        <TextBlock VerticalAlignment="Center" Margin="10" Grid.Row="0" Grid.Column="0">First Name:</TextBlock>
        <TextBox Margin="10" Grid.Row="0" Grid.Column="1" Text="{Binding FirstName, Mode=TwoWay}"/>

        <TextBlock VerticalAlignment="Center" Margin="10" Grid.Row="1" Grid.Column="0">Last Name:</TextBlock>
        <TextBox Margin="10" Grid.Row="1" Grid.Column="1" Text="{Binding LastName}"/>

        <TextBlock VerticalAlignment="Center" Margin="10" Grid.Row="2" Grid.Column="0">Address:</TextBlock>
        <TextBox Margin="10" Grid.Row="2" Grid.Column="1" Text="{Binding Address}"/>

    </Grid>

    <Border Background="Tan" Margin="10">
        <TextBlock x:Name="CheckFirstName"/>
    </Border>

</StackPanel>

Code behind:

public Page()
{
    InitializeComponent();

    Customer customer = new Customer();
    customer.FirstName = "Jim";
    customer.LastName = "Taylor";
    customer.Address = "72384 South Northern Blvd.";

    GridCustomerDetails.DataContext = customer;

    Customer customerOutput = (Customer)GridCustomerDetails.DataContext;
    CheckFirstName.Text = customer.FirstName;

}