CheckBox binding working only 1 way

I have a Datagrid in whitch I have a column who has a CheckBox inside (I know that DataGridCheckBoxColumn exist but I don't like the behavior (clicking once to select and a second time to alter the value)).

This checkbox is binded this way

<DataGridTemplateColumn x:Name="checkColumn" Header="Check">
    <DataGridTemplateColumn.CellTemplate>
       <DataTemplate>
          <CheckBox IsChecked="{Binding load, Mode=TwoWay}" />
       </DataTemplate>
    </DataGridTemplateColumn.CellTemplate>
</DataGridTemplateColumn>

The TwoWay mode is supposed to be default but I added it anyway since it seems to work as if it was read only.

And the property load is simply

private bool _load;
public bool load
{
   get { return _load; }
   set { _load = value;}
}

But in the end the binding doesn't work properly. For each item I have the correct value displayed by default (checked/not checked), but if I check/uncheck the checkboxes, nothing changes on the data (but the checkboxes do change visually, but the setter of the item is never called).

I suspect that the problem is not a big one, since the default values are read, the binding is not completely wrong, but I can't find out why the setter is never called.